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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-tr 4901 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5662). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4902 (which is suggestive of the word "transitive"), dftr3 4904, dftr4 4905, dftr5 4903, and (when 𝐴 is a set) unisuc 5958. 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 4902* 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 4903* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)

Theoremdftr3 4904* 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 4905 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremtreq 4906 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Theoremtrel 4907 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 4908 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))

Theoremtrss 4909 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 4910 Obsolete proof of trss 4909 as of 26-Jul-2021. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))

Theoremtrin 4911 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))

Theoremtr0 4912 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅

Theoremtrv 4913 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V

Theoremtriun 4914* 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 4915* 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 4916* 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 4917 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 4918 Obsolete version of trintss 4917 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 4919* 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 6133). 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 4924, where you can find some further remarks. A slightly more compact version is shown as axrep2 4921. A quite different variant is zfrep6 7295, which if used in place of ax-rep 4919 would also require that the Separation Scheme axsep 4928 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 8923 and the Boundedness Axiom bnd 8924.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4928, Null Set axnul 4936, and Pairing axpr 5050, 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 4929, ax-nul 4937, and ax-pr 5051 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

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

Theoremaxrep1 4920* 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 4919 axrep1 4920 axrep2 4921 axrepnd 9604 zfcndrep 9624 = ax-rep 4919. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theoremaxrep2 4921* 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 4922* Axiom of Replacement slightly strengthened from axrep2 4921; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theoremaxrep4 4923* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremaxrep5 4924* 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 4925* 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 4926* 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 4927* 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 4928* 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 4919. 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 3571. 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 4931, 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 4985 shows (contradicting zfauscl 4931). However, as axsep2 4930 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 4929 from ax-rep 4919.

This theorem should not be referenced by any proof. Instead, use ax-sep 4929 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 4929* The Axiom of Separation of ZF set theory. See axsep 4928 for more information. It was derived as axsep 4928 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 4930* A less restrictive version of the Separation Scheme axsep 4928, 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 4929 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremzfauscl 4931* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4929, we invoke the Axiom of Extensionality (indirectly via vtocl 3395), 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 4985 shows. (Contributed by NM, 21-Jun-1993.)

𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theorembm1.3ii 4932* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4929. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theoremax6vsep 4933* Derive ax6v 2051 (a weakened version of ax-6 2050 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 4929 and Extensionality ax-ext 2736. See ax6 2392 for the derivation of ax-6 2050 from ax6v 2051. (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 4934* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2741 to strengthen the hypothesis in the form of axnul 4936). (Contributed by NM, 22-Dec-2007.)
𝑥𝑦 ¬ 𝑦𝑥       ∃!𝑥𝑦 ¬ 𝑦𝑥

TheoremaxnulALT 4935* Alternate proof of axnul 4936, proved from propositional calculus, ax-gen 1867, ax-4 1882, sp 2196, and ax-rep 4919. To check this, replace sp 2196 with the obsolete axiom ax-c5 34668 in the proof of axnulALT 4935 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 4936* 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 4929. 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 4934).

This proof, suggested by Jeff Hoffman, uses only ax-4 1882 and ax-gen 1867 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 4929 implies the existence of at least one set. Note that Kunen's version of ax-sep 4929 (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 4935 for a proof directly from ax-rep 4919.

This theorem should not be referenced by any proof. Instead, use ax-nul 4937 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 4937* The Null Set Axiom of ZF set theory. It was derived as axnul 4936 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 4938 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 4937. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V

TheoremsseliALT 4939 Alternate proof of sseli 3736 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3737. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)

Theoremcsbexg 4940 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 4941 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 4942 A version of unisn 4599 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
{𝐴} ∈ {∅, 𝐴}

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4943* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ ∃𝑥𝑦 𝑦𝑥

Theoremvprc 4944 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 4945 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴

Theoremvnex 4946 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V

Theoreminex1 4947 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoreminex2 4948 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V

Theoreminex1g 4949 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremssex 4950 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 4929 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theoremssexi 4951 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 4952 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 4953 A subclass of a set is a set. Deduction form of ssexg 4952. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremprcssprc 4954 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Theoremsselpwd 4955 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremdifexg 4956 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremdifexi 4957 Existence of a difference, inference version of difexg 4956. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremzfausab 4958* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 4959* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 4960* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremrabexd 4961* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4962. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)

Theoremrabex2 4962* 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 4963* 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

Theoremelssabg 4964* Membership in a class abstraction involving a subset. Unlike elabg 3487, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoremintex 4965 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 4966 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)

Theoremintexab 4967 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrab 4968 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexg 4969* 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 4970* 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 4971* 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 4972 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 4973 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwi2 4974 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theorempwnss 4975 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwne 4976 No set equals its power set. The sethood antecedent is necessary; compare pwv 4581. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(𝐴𝑉 → 𝒫 𝐴𝐴)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 4977* Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)
{𝑥𝐴𝐴 ∈ V} ∈ V

Theoremclass2seteq 4978* Equality theorem based on class2set 4977. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)

Theorem0elpw 4979 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴

Theorempwne0 4980 A power class is never empty. (Contributed by NM, 3-Sep-2018.)
𝒫 𝐴 ≠ ∅

Theorem0nep0 4981 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}

Theorem0inp0 4982 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
(𝐴 = ∅ → ¬ 𝐴 = {∅})

Theoremunidif0 4983 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴

Theoremiin0 4984* An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
(𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)

Theoremnotzfaus 4985* In the Separation Scheme zfauscl 4931, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theoremintv 4986 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅

Theoremaxpweq 4987* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4988 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
𝐴 ∈ V       (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))

2.3  ZF Set Theory - add the Axiom of Power Sets

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 4988* Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the power set of a given set 𝑥 i.e. contains every subset of 𝑥. The variant axpow2 4990 uses explicit subset notation. A version using class notation is pwex 4993. (Contributed by NM, 21-Jun-1993.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfpow 4989* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theoremaxpow2 4990* A variant of the Axiom of Power Sets ax-pow 4988 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremaxpow3 4991* A variant of the Axiom of Power Sets ax-pow 4988. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremel 4992* Every set is an element of some other set. See elALT 5055 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑦 𝑥𝑦

Theorempwex 4993 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴 ∈ V       𝒫 𝐴 ∈ V

Theoremvpwex 4994 The powerset of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝒫 𝑥 ∈ V

Theorempwexg 4995 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
(𝐴𝑉 → 𝒫 𝐴 ∈ V)

Theoremabssexg 4996* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)

TheoremsnexALT 4997 Alternate proof of snex 5053 using Power Set (ax-pow 4988) instead of Pairing (ax-pr 5051). Unlike in the proof of zfpair 5049, Replacement (ax-rep 4919) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝐴} ∈ V

Theoremp0ex 4998 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4999. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V

Theoremp0exALT 4999 Alternate proof of p0ex 4998 which is quite different and longer if snexALT 4997 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
{∅} ∈ V

Theorempp0ex 5000 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
{∅, {∅}} ∈ V

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