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Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpteq12f 4701 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12dva 4702* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12dv 4703* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12d 4704 An equality inference for the maps to notation. Compare mpteq12dv 4703. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12df 4705 An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12 4706* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)
((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq1 4707* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmpteq1d 4708* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmpteq1i 4709* An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴 = 𝐵       (𝑥𝐴𝐶) = (𝑥𝐵𝐶)

Theoremmpteq2ia 4710 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝑥𝐴𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Theoremmpteq2i 4711 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
𝐵 = 𝐶       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Theoremmpteq12i 4712 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝑥𝐴𝐵) = (𝑥𝐶𝐷)

Theoremmpteq2da 4713 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremmpteq2dva 4714* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremmpteq2dv 4715* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremnfmpt 4716* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝑦𝐴𝐵)

Theoremnfmpt1 4717 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
𝑥(𝑥𝐴𝐵)

Theoremcbvmptf 4718* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmpt 4719* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmptv 4720* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremmptv 4721* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}

2.1.24  Transitive classes

Syntaxwtr 4722 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴

Definitiondf-tr 4723 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5477). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4724 (which is suggestive of the word "transitive"), dftr3 4726, dftr4 4727, dftr5 4725, and (when 𝐴 is a set) unisuc 5770. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 𝐴𝐴)

Theoremdftr2 4724* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
(Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))

Theoremdftr5 4725* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)

Theoremdftr3 4726* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)

Theoremdftr4 4727 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremtreq 4728 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Theoremtrel 4729 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))

Theoremtrel3 4730 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))

Theoremtrss 4731 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))

TheoremtrssOLD 4732 Obsolete proof of trss 4731 as of 26-Jul-2021. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))

Theoremtrin 4733 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))

Theoremtr0 4734 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅

Theoremtrv 4735 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V

Theoremtriun 4736* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Theoremtruni 4737* The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrint 4738* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrintss 4739 Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)
((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)

TheoremtrintssOLD 4740 Obsolete version of trintss 4739 as of 30-Oct-2021. (Contributed by Scott Fenton, 3-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)

2.2  ZF Set Theory - add the Axiom of Replacement

2.2.1  Introduce the Axiom of Replacement

Axiomax-rep 4741* Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 5944). Although 𝜑 may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and 𝜑 encodes the predicate "the value of the function at 𝑤 is 𝑧." Thus, 𝜑 will ordinarily have free variables 𝑤 and 𝑧- think of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the quantifier 𝑦 in order to "protect" the axiom from any 𝜑 containing 𝑦, thus allowing us to eliminate any restrictions on 𝜑. Another common variant is derived as axrep5 4746, where you can find some further remarks. A slightly more compact version is shown as axrep2 4743. A quite different variant is zfrep6 7096, which if used in place of ax-rep 4741 would also require that the Separation Scheme axsep 4750 be stated as a separate axiom.

There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 8714 and the Boundedness Axiom bnd 8715.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4750, Null Set axnul 4758, and Pairing axpr 4876, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 4751, ax-nul 4759, and ax-pr 4877 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))

Theoremaxrep1 4742* The version of the Axiom of Replacement used in the Metamath Solitaire applet http://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 4741 axrep1 4742 axrep2 4743 axrepnd 9376 zfcndrep 9396 = ax-rep 4741. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theoremaxrep2 4743* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))

Theoremaxrep3 4744* Axiom of Replacement slightly strengthened from axrep2 4743; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theoremaxrep4 4745* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremaxrep5 4746* Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremzfrepclf 4747* An inference rule based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
𝑥𝐴    &   𝐴 ∈ V    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremzfrep3cl 4748* An inference rule based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
𝐴 ∈ V    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremzfrep4 4749* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
{𝑥𝜑} ∈ V    &   (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))       {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V

2.2.2  Derive the Axiom of Separation

Theoremaxsep 4750* Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. As we show here, it is redundant if we assume Replacement in the form of ax-rep 4741. Some textbooks present Separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger Replacement. The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 𝑥𝑧) so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3421. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

The variable 𝑥 can appear free in the wff 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable requirement (\$d) that 𝑥 not appear in 𝜑.

For a version using a class variable, see zfauscl 4753, which requires the Axiom of Extensionality as well as Separation for its derivation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 4810 shows (contradicting zfauscl 4753). However, as axsep2 4752 shows, we can eliminate the restriction that 𝑧 not occur in 𝜑.

Note: the distinct variable restriction that 𝑧 not occur in 𝜑 is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 4751 from ax-rep 4741.

This theorem should not be referenced by any proof. Instead, use ax-sep 4751 below so that the uses of the Axiom of Separation can be more easily identified. (Contributed by NM, 11-Sep-2006.) (New usage is discouraged.)

𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Axiomax-sep 4751* The Axiom of Separation of ZF set theory. See axsep 4750 for more information. It was derived as axsep 4750 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 11-Sep-2006.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremaxsep2 4752* A less restrictive version of the Separation Scheme axsep 4750, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 4751 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremzfauscl 4753* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4751, we invoke the Axiom of Extensionality (indirectly via vtocl 3249), which is needed for the justification of class variable notation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 4810 shows. (Contributed by NM, 21-Jun-1993.)

𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theorembm1.3ii 4754* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4751. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theoremax6vsep 4755* Derive ax6v 1886 (a weakened version of ax-6 1885 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 4751 and Extensionality ax-ext 2601. See ax6 2250 for the derivation of ax-6 1885 from ax6v 1886. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

2.2.3  Derive the Null Set Axiom

Theoremzfnuleu 4756* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2606 to strengthen the hypothesis in the form of axnul 4758). (Contributed by NM, 22-Dec-2007.)
𝑥𝑦 ¬ 𝑦𝑥       ∃!𝑥𝑦 ¬ 𝑦𝑥

TheoremaxnulALT 4757* Alternate proof of axnul 4758, proved from propositional calculus, ax-gen 1719, ax-4 1734, sp 2051, and ax-rep 4741. To check this, replace sp 2051 with the obsolete axiom ax-c5 33687 in the proof of axnulALT 4757 and type the Metamath command 'SHOW TRACEBACK axnulALT / AXIOMS'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥

Theoremaxnul 4758* The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 4751. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4756).

This proof, suggested by Jeff Hoffman, uses only ax-4 1734 and ax-gen 1719 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus, our ax-sep 4751 implies the existence of at least one set. Note that Kunen's version of ax-sep 4751 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed i.e. prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating 𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10).

See axnulALT 4757 for a proof directly from ax-rep 4741.

This theorem should not be referenced by any proof. Instead, use ax-nul 4759 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

𝑥𝑦 ¬ 𝑦𝑥

Axiomax-nul 4759* The Null Set Axiom of ZF set theory. It was derived as axnul 4758 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.)
𝑥𝑦 ¬ 𝑦𝑥

Theorem0ex 4760 The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 4759. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V

TheoremsseliALT 4761 Alternate proof of sseli 3584 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3585. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)

Theoremcsbexg 4762 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
(∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)

Theoremcsbex 4763 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by NM, 17-Aug-2018.)
𝐵 ∈ V       𝐴 / 𝑥𝐵 ∈ V

Theoremunisn2 4764 A version of unisn 4424 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
{𝐴} ∈ {∅, 𝐴}

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4765* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ ∃𝑥𝑦 𝑦𝑥

Theoremvprc 4766 The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)
¬ V ∈ V

Theoremnvel 4767 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴

Theoremvnex 4768 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V

Theoreminex1 4769 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoreminex2 4770 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V

Theoreminex1g 4771 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremssex 4772 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4751 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theoremssexi 4773 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 4774 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theoremssexd 4775 A subclass of a set is a set. Deduction form of ssexg 4774. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremprcssprc 4776 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Theoremsselpwd 4777 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremdifexg 4778 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremdifexi 4779 Existence of a difference, inference version of difexg 4778. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

TheoremdifexOLD 4780 Obsolete version of difexi 4779 as of 26-Mar-2021. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝑉       (𝐴𝐵) ∈ V

Theoremzfausab 4781* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 4782* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 4783* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremrabexd 4784* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4785. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)

Theoremrabex2 4785* Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴 ∈ V       𝐵 ∈ V

Theoremrab2ex 4786* A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑦𝐴𝜓}    &   𝐴 ∈ V       {𝑥𝐵𝜑} ∈ V

Theoremrabex2OLD 4787* Obsolete version of rabex2 4785 as of 26-Mar-2021. (Contributed by AV, 16-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴𝑉       𝐵 ∈ V

Theoremrab2exOLD 4788* Obsolete version of rab2ex 4786 as of 26-Mar-2021. (Contributed by AV, 16-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = {𝑦𝐴𝜓}    &   𝐴𝑉       {𝑥𝐵𝜑} ∈ V

Theoremelssabg 4789* Membership in a class abstraction involving a subset. Unlike elabg 3339, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoremintex 4790 The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)
(𝐴 ≠ ∅ ↔ 𝐴 ∈ V)

Theoremintnex 4791 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)

Theoremintexab 4792 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrab 4793 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexg 4794* The existence of a class intersection. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by FL, 19-Sep-2011.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)

Theoremintabs 4795* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = {𝑦𝜓} → (𝜑𝜒))    &   ( {𝑦𝜓} ⊆ 𝐴𝜒)        {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}

Theoreminuni 4796* The intersection of a union 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)
( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}

Theoremelpw2g 4797 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 4798 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwi2 4799 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theorempwnss 4800 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

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