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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuc11reg 8701 The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
(suc 𝐴 = suc 𝐵𝐴 = 𝐵)
 
Theoremdford2 8702* Assuming ax-reg 8674, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
 
2.5.2  Axiom of Infinity equivalents
 
Theoreminf0 8703* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 8720. (Contributed by NM, 15-Oct-1996.)
ω ∈ V       𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
 
Theoreminf1 8704 Variation of Axiom of Infinity (using zfinf 8721 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
 
Theoreminf2 8705* Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8721 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
 
Theoreminf3lema 8706* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
 
Theoreminf3lemb 8707* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐹‘∅) = ∅
 
Theoreminf3lemc 8708* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹𝐴)))
 
Theoreminf3lemd 8709* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)
 
Theoreminf3lem1 8710* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))
 
Theoreminf3lem2 8711* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ 𝑥))
 
Theoreminf3lem3 8712* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8677. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ (𝐹‘suc 𝐴)))
 
Theoreminf3lem4 8713* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ⊊ (𝐹‘suc 𝐴)))
 
Theoreminf3lem5 8714* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝐹𝐵) ⊊ (𝐹𝐴)))
 
Theoreminf3lem6 8715* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
 
Theoreminf3lem7 8716* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8717 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7304. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
 
Theoreminf3 8717 Our Axiom of Infinity ax-inf 8720 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8705, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 8722 and zfinf2 8724.) The main proof is provided by inf3lema 8706 through inf3lem7 8716, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8716. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.
       (As posted to sci.logic on 30-Oct-1996, with annotations added.)

       Theorem:  The statement "There exists a nonempty set that is a subset
       of its union" implies the Axiom of Infinity.

       Proof:  Let X be a nonempty set which is a subset of its union; the
       latter
       property is equivalent to saying that for any y in X, there exists a z
       in X
       such that y is in z.

       Define by finite recursion a function F:omega-->(power X) such that
       F_0 = 0  (See inf3lemb 8707.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8708.)
       Note: ^ means intersect, < means \in ("element of").
       (Finite recursion as typically done requires the existence of omega;
       to avoid this we can just use transfinite recursion restricted to omega.
       F is a class-term that is not necessarily a set at this point.)

       Lemma 1.  F_n subset F_n+1.  (See inf3lem1 8710.)
       Proof:  By induction:  F_0 subset F_1.  If y < F_n+1, then y^X subset
       F_n,
       so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2.

       Lemma 2.  F_n =/= X.  (See inf3lem2 8711.)
       Proof:  By induction:  F_0 =/= X because X is not empty.  Assume F_n =/=
       X.
       Then there is a y in X that is not in F_n.  By definition of X, there is
       a
       z in X that contains y.  Suppose F_n+1 = X.  Then z is in F_n+1, and z^X
       contains y, so z^X is not a subset of F_n, contrary to the definition of
       F_n+1.

       Lemma 3.  F_n =/= F_n+1.  (See inf3lem3 8712.)
       Proof:  Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have
       F_n+1 = {y<X | y^(X-F_n) = 0}.  Let q = {y<X-F_n | y^(X-F_n) = 0}.
       Then q subset F_n+1.  Since X-F_n is not empty by Lemma 2 and q is the
       set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q
       and therefore F_n+1 have an element not in F_n.

       Lemma 4.  F_n proper_subset F_n+1.  (See inf3lem4 8713.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8714.)
       Proof:  Fix m and use induction on n > m.  Basis: F_m proper_subset
       F_m+1
       by Lemma 4.  Induction:  Assume F_m proper_subset F_n.  Then since F_n
       proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper
       subset.

       By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1.  (See inf3lem6 8715.)
       Thus, the inverse of F is a function with range omega and domain a
       subset
       of power X, so omega exists by Replacement.  (See inf3lem7 8716.)
       Q.E.D.
       
(Contributed by NM, 29-Oct-1996.)
𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)       ω ∈ V
 
Theoreminfeq5i 8718 Half of infeq5 8719. (Contributed by Mario Carneiro, 16-Nov-2014.)
(ω ∈ V → ∃𝑥 𝑥 𝑥)
 
Theoreminfeq5 8719 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 8725.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
(∃𝑥 𝑥 𝑥 ↔ ω ∈ V)
 
2.6  ZF Set Theory - add the Axiom of Infinity
 
2.6.1  Introduce the Axiom of Infinity
 
Axiomax-inf 8720* Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set 𝑥, an infinite set 𝑦 built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 8704 and inf2 8705). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8724 and omex 8725 and are based on the (nontrivial) proof of inf3 8717. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8723. Theorem inf0 8703 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8727 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8723 requires this axiom along with Regularity ax-reg 8674 for its derivation (as theorem axinf2 8722 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8723 instead of this one. The derivation of this axiom from ax-inf2 8723 is shown by theorem axinf 8726.

Proofs should normally use the standard version ax-inf2 8723 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
 
Theoremzfinf 8721* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
 
Theoremaxinf2 8722* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8720 and Regularity ax-reg 8674.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8723 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
 
Axiomax-inf2 8723* A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 8724 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8722 above, using our version of Infinity ax-inf 8720 and the Axiom of Regularity ax-reg 8674. We will reference ax-inf2 8723 instead of axinf2 8722 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8720 from ax-inf2 8723 is shown by theorem axinf 8726. (Contributed by NM, 3-Nov-1996.)
𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
 
Theoremzfinf2 8724* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8723 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
 
2.6.2  Existence of omega (the set of natural numbers)
 
Theoremomex 8725 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 8703.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 7244 and Fin = V (the universe of all sets) by fineqv 8352. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7253 through peano5 7257 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

ω ∈ V
 
Theoremaxinf 8726* The first version of the Axiom of Infinity ax-inf 8720 proved from the second version ax-inf2 8723. Note that we didn't use ax-reg 8674, unlike the other direction axinf2 8722. (Contributed by NM, 24-Apr-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
 
Theoreminf5 8727 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 8719). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)
𝑥 𝑥 𝑥
 
Theoremomelon 8728 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
ω ∈ On
 
Theoremdfom3 8729* The class of natural numbers omega can be defined as the smallest "inductive set," which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.)
ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
 
Theoremelom3 8730* A simplification of elom 7236 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
(𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥𝐴𝑥))
 
Theoremdfom4 8731* A simplification of df-om 7234 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
 
Theoremdfom5 8732 ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
ω = {𝑥 ∣ Lim 𝑥}
 
Theoremoancom 8733 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
(1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)
 
Theoremisfinite 8734 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)
(𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
 
Theoremfict 8735 A finite set is countable (weaker version of isfinite 8734). (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝐴 ∈ Fin → 𝐴 ≼ ω)
 
Theoremnnsdom 8736 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)
(𝐴 ∈ ω → 𝐴 ≺ ω)
 
Theoremomenps 8737 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)
ω ≈ (ω ∖ {∅})
 
Theoremomensuc 8738 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
ω ≈ suc ω
 
Theoreminfdifsn 8739 Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
(ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
 
Theoreminfdiffi 8740 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)
 
Theoremunbnn3 8741* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 8393 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)
 
Theoremnoinfep 8742* Using the Axiom of Regularity in the form zfregfr 8686, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)
 
2.6.3  Cantor normal form
 
Syntaxccnf 8743 Extend class notation with the Cantor normal form function.
class CNF
 
Definitiondf-cnf 8744* Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥𝑜 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑𝑜 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 8777 of this function in terms of df-oi 8592. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
 
Theoremcantnffval 8745* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
 
Theoremcantnfdm 8746* The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
 
Theoremcantnfvalf 8747* Lemma for cantnf 8775. The function appearing in cantnfval 8750 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
𝐹 = seq𝜔((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)), ∅)       𝐹:ω⟶On
 
Theoremcantnfs 8748 Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
 
Theoremcantnfcl 8749 Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)       (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
 
Theoremcantnfval 8750* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
 
Theoremcantnfval2 8751* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))
 
Theoremcantnfsuc 8752* The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))
 
Theoremcantnfle 8753* A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴𝑜 𝑥) ·𝑜 (𝐹𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)    &   (𝜑𝐶𝐵)       (𝜑 → ((𝐴𝑜 𝐶) ·𝑜 (𝐹𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
 
Theoremcantnflt 8754* An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴𝑜 𝐶 where 𝐶 is larger than any exponent (𝐺𝑥), 𝑥𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   (𝜑𝐾 ∈ suc dom 𝐺)    &   (𝜑𝐶 ∈ On)    &   (𝜑 → (𝐺𝐾) ⊆ 𝐶)       (𝜑 → (𝐻𝐾) ∈ (𝐴𝑜 𝐶))
 
Theoremcantnflt2 8755 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐹𝑆)    &   (𝜑 → ∅ ∈ 𝐴)    &   (𝜑𝐶 ∈ On)    &   (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴𝑜 𝐶))
 
Theoremcantnff 8756 The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴𝑜 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
 
Theoremcantnf0 8757 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑 → ∅ ∈ 𝐴)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
 
Theoremcantnfrescl 8758* A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑𝐵𝐷)    &   ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   𝑇 = dom (𝐴 CNF 𝐷)       (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
 
Theoremcantnfres 8759* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑𝐵𝐷)    &   ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   𝑇 = dom (𝐴 CNF 𝐷)    &   (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
 
Theoremcantnfp1lem1 8760* Lemma for cantnfp1 8763. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑𝐹𝑆)
 
Theoremcantnfp1lem2 8761* Lemma for cantnfp1 8763. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))    &   (𝜑 → ∅ ∈ 𝑌)    &   𝑂 = OrdIso( E , (𝐹 supp ∅))       (𝜑 → dom 𝑂 = suc dom 𝑂)
 
Theoremcantnfp1lem3 8762* Lemma for cantnfp1 8763. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))    &   (𝜑 → ∅ ∈ 𝑌)    &   𝑂 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐹‘(𝑂𝑘))) +𝑜 𝑧)), ∅)    &   𝐾 = OrdIso( E , (𝐺 supp ∅))    &   𝑀 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐾𝑘)) ·𝑜 (𝐺‘(𝐾𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
 
Theoremcantnfp1 8763* If 𝐹 is created by adding a single term (𝐹𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
 
Theoremoemapso 8764* The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 8635). (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑𝑇 Or 𝑆)
 
Theoremoemapval 8765* Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)       (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
 
Theoremoemapvali 8766* If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}       (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
 
Theoremcantnflem1a 8767* Lemma for cantnf 8775. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}       (𝜑𝑋 ∈ (𝐺 supp ∅))
 
Theoremcantnflem1b 8768* Lemma for cantnf 8775. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))       ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
 
Theoremcantnflem1c 8769* Lemma for cantnf 8775. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))       ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
 
Theoremcantnflem1d 8770* Lemma for cantnf 8775. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))
 
Theoremcantnflem1 8771* Lemma for cantnf 8775. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺))
 
Theoremcantnflem2 8772* Lemma for cantnf 8775. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)       (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
 
Theoremcantnflem3 8773* Lemma for cantnf 8775. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 7858 to factor 𝐶 into the form ((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴𝑜 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴𝑜 𝑋) ≤ (𝐴𝑜 𝑋) ·𝑜 𝑌𝐶, 𝑍 has a normal form, and by appending the term (𝐴𝑜 𝑋) ·𝑜 𝑌 using cantnfp1 8763 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)    &   𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}    &   𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)    &   (𝜑𝐺𝑆)    &   (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
 
Theoremcantnflem4 8774* Lemma for cantnf 8775. Complete the induction step of cantnflem3 8773. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)    &   𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}    &   𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)       (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
 
Theoremcantnf 8775* The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴𝑜 𝑓(𝑎1)) ∘ 𝑎1) +𝑜 ((𝐴𝑜 𝑓(𝑎2)) ∘ 𝑎2) +𝑜 ... over all indices 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴𝑜 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 8759, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))
 
Theoremoemapwe 8776* The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴𝑜 𝐵)))
 
Theoremcantnffval2 8777* An alternate definition of df-cnf 8744 which relies on cantnf 8775. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8746 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
 
Theoremcantnff1o 8778 Simplify the isomorphism of cantnf 8775 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))
 
Theoremwemapwe 8779* Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑆 We 𝐵)    &   (𝜑𝐵 ≠ ∅)    &   𝐹 = OrdIso(𝑅, 𝐴)    &   𝐺 = OrdIso(𝑆, 𝐵)    &   𝑍 = (𝐺‘∅)       (𝜑𝑇 We 𝑈)
 
Theoremoef1o 8780* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6719.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
(𝜑𝐹:𝐴1-1-onto𝐶)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ (On ∖ 1𝑜))    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐶 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑 → (𝐹‘∅) = ∅)    &   𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))    &   𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))       (𝜑𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
 
Theoremcnfcomlem 8781* Lemma for cnfcom 8782. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   (𝜑𝐼 ∈ dom 𝐺)    &   (𝜑𝑂 ∈ (ω ↑𝑜 (𝐺𝐼)))    &   (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)       (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
 
Theoremcnfcom 8782* Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   (𝜑𝐼 ∈ dom 𝐺)       (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
 
Theoremcnfcom2lem 8783* Lemma for cnfcom2 8784. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ∅ ∈ 𝐵)       (𝜑 → dom 𝐺 = suc dom 𝐺)
 
Theoremcnfcom2 8784* Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ∅ ∈ 𝐵)       (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
 
Theoremcnfcom3lem 8785* Lemma for cnfcom3 8786. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ω ⊆ 𝐵)       (𝜑𝑊 ∈ (On ∖ 1𝑜))
 
Theoremcnfcom3 8786* Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 8788.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ω ⊆ 𝐵)    &   𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))    &   𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))    &   𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))       (𝜑𝑁:𝐵1-1-onto→(ω ↑𝑜 𝑊))
 
Theoremcnfcom3clem 8787* Lemma for cnfcom3c 8788. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   𝐹 = ((ω CNF 𝐴)‘𝑏)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))    &   𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))    &   𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))    &   𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)       (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
 
Theoremcnfcom3c 8788* Wrap the construction of cnfcom3 8786 into an existence quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
(𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
 
2.6.4  Transitive closure
 
Theoremtrcl 8789* For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 8790 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐴 ∈ V    &   𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)    &   𝐶 = 𝑦 ∈ ω (𝐹𝑦)       (𝐴𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥))
 
Theoremtz9.1 8790* Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8789 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself?

(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)

𝐴 ∈ V       𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))
 
Theoremtz9.1c 8791* Alternate expression for the existence of transitive closures tz9.1 8790: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V        {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V
 
Theoremepfrs 8792* The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 8793. (Contributed by Mario Carneiro, 22-Mar-2013.)
(( E Fr 𝐴𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
 
Theoremzfregs 8793* The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 8792. (Contributed by NM, 17-Sep-2003.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)
 
Theoremzfregs2 8794* Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))
 
Theoremsetind 8795* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
 
Theoremsetind2 8796 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
(𝒫 𝐴𝐴𝐴 = V)
 
Syntaxctc 8797 Extend class notation to include the transitive closure function.
class TC
 
Definitiondf-tc 8798* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
TC = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ Tr 𝑦)})
 
Theoremtcvalg 8799* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 8720; see tz9.1 8790.) (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
 
Theoremtcid 8800 Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉𝐴 ⊆ (TC‘𝐴))
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