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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtruni 5001* 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)

Theoremtriin 5002 An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Theoremtrint 5003* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

TheoremtrintOLD 5004* Obsolete proof of trintOLD 5004 as of 3-Oct-2022. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrintss 5005 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)

2.2  ZF Set Theory - add the Axiom of Replacement

2.2.1  Introduce the Axiom of Replacement

Axiomax-rep 5006* 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 6221). 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 5012, where you can find some further remarks. A slightly more compact version is shown as axrep2 5009. A quite different variant is zfrep6 7413, which if used in place of ax-rep 5006 would also require that the Separation Scheme axsep 5016 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 9051 and the Boundedness Axiom bnd 9052.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5016, Null Set axnul 5024, and Pairing axpr 5137, 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 5017, ax-nul 5025, and ax-pr 5138 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

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

Theoremaxrep1 5007* 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 5006 axrep1 5007 axrep2 5009 axrepnd 9751 zfcndrep 9771 = ax-rep 5006. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theoremaxreplem 5008* Lemma for axrep2 5009 and axrep3 5010. (Contributed by BJ, 6-Aug-2022.)
(𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))

Theoremaxrep2 5009* 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 5010* Axiom of Replacement slightly strengthened from axrep2 5009; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theoremaxrep4 5011* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremaxrep5 5012* 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 5013* An inference 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 5014* An inference 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 5015* 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 5016* 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 5006. 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 3651. 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 5019, 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 5074 shows (showing the necessity of that condition in zfauscl 5019). However, as axsep2 5018 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 5017 from ax-rep 5006.

This theorem should not be referenced by any proof. Instead, use ax-sep 5017 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 5017* The Axiom of Separation of ZF set theory. See axsep 5016 for more information. It was derived as axsep 5016 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 5018* A less restrictive version of the Separation Scheme axsep 5016, 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 5017 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremzfauscl 5019* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5017, we invoke the Axiom of Extensionality (indirectly via vtocl 3460), 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 5074 shows. (Contributed by NM, 21-Jun-1993.)

𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theorembm1.3ii 5020* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5017. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theoremax6vsep 5021* Derive ax6v 2022 (a weakened version of ax-6 2021 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5017 and Extensionality ax-ext 2754. See ax6 2348 for the derivation of ax-6 2021 from ax6v 2022. (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

TheoremzfnuleuOLD 5022* Obsolete version of nulmo 2761 as of 17-Sep-2022. (Contributed by NM, 22-Dec-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥       ∃!𝑥𝑦 ¬ 𝑦𝑥

TheoremaxnulALT 5023* Alternate proof of axnul 5024, proved from propositional calculus, ax-gen 1839, ax-4 1853, sp 2167, and ax-rep 5006. To check this, replace sp 2167 with the obsolete axiom ax-c5 35037 in the proof of axnulALT 5023 and type the Metamath program "MM> SHOW TRACEBACK axnulALT / AXIOMS" command. (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 5024* 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 5017. 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 nulmo 2761).

This proof, suggested by Jeff Hoffman, uses only ax-4 1853 and ax-gen 1839 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 5017 implies the existence of at least one set. Note that Kunen's version of ax-sep 5017 (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 5023 for a proof directly from ax-rep 5006.

This theorem should not be referenced by any proof. Instead, use ax-nul 5025 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 5025* The Null Set Axiom of ZF set theory. It was derived as axnul 5024 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 5026 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 5025. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V

Theoremal0ssb 5027* The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)
(∀𝑦 𝑋𝑦𝑋 = ∅)

TheoremsseliALT 5028 Alternate proof of sseli 3817 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3818. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)

Theoremcsbexg 5029 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 5030 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 5031 A version of unisn 4687 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
{𝐴} ∈ {∅, 𝐴}

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 5032* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ ∃𝑥𝑦 𝑦𝑥

Theoremvnex 5033 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V

Theoremvprc 5034 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 5035 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴

Theoreminex1 5036 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoreminex2 5037 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V

Theoreminex1g 5038 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremssex 5039 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 5017 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theoremssexi 5040 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 5041 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 5042 A subclass of a set is a set. Deduction form of ssexg 5041. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremprcssprc 5043 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Theoremsselpwd 5044 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremdifexg 5045 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremdifexi 5046 Existence of a difference, inference version of difexg 5045. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremzfausab 5047* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 5048* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 5049* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremrabexd 5050* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5051. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)

Theoremrabex2 5051* 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 5052* 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 5053* Membership in a class abstraction involving a subset. Unlike elabg 3556, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoremintex 5054 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 5055 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)

Theoremintexab 5056 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrab 5057 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexg 5058* 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 5059* 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 5060* 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 5061 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 5062 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwi2 5063 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theorempwnss 5064 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwne 5065 No set equals its power set. The sethood antecedent is necessary; compare pwv 4668. (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 5066* 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 5067* Equality theorem based on class2set 5066. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)

Theorem0elpw 5068 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴

Theorempwne0 5069 A power class is never empty. (Contributed by NM, 3-Sep-2018.)
𝒫 𝐴 ≠ ∅

Theorem0nep0 5070 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}

Theorem0inp0 5071 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 5072 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴

Theoremiin0 5073* An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
(𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)

Theoremnotzfaus 5074* In the Separation Scheme zfauscl 5019, 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 if that requirement is not met. (Contributed by NM, 8-Feb-2006.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theoremintv 5075 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅

Theoremaxpweq 5076* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5077 is not used by the proof. When ax-pow 5077 is assumed and 𝐴 is a set, both sides of the biconditional hold. In ZF, both sides hold if and only if 𝐴 is a set (see pwexr 7251). (Contributed by NM, 22-Jun-2009.)
(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 5077* 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 5079 uses explicit subset notation. A version using class notation is pwex 5092. (Contributed by NM, 21-Jun-1993.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfpow 5078* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theoremaxpow2 5079* A variant of the Axiom of Power Sets ax-pow 5077 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremaxpow3 5080* A variant of the Axiom of Power Sets ax-pow 5077. 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 5081* Every set is an element of some other set. See elALT 5142 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑦 𝑥𝑦

Theoremdtru 5082* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both 𝑥 and 𝑦 (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 2075.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2754 or ax-sep 5017. See dtruALT 5099 for a shorter proof using these axioms.

The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.)

¬ ∀𝑥 𝑥 = 𝑦

Theoremdtrucor 5083* Corollary of dtru 5082. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 5084. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦

Theoremdtrucor2 5084 The theorem form of the deduction dtrucor 5083 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

Theoremdvdemo1 5085* Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑦 to be distinct, but no others. It bundles the theorem schemes 𝑥(𝑥 = 𝑦𝑥𝑥) and 𝑥(𝑥 = 𝑦𝑦𝑥). Compare dvdemo2 5086. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)

Theoremdvdemo2 5086* Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑧 to be distinct, but no others. It bundles the theorem schemes 𝑥(𝑥 = 𝑥𝑧𝑥) and 𝑥(𝑥 = 𝑦𝑦𝑥). Compare dvdemo1 5085. (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)

Theoremnfnid 5087 A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtru 5082 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥

Theoremnfcvb 5088 The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then 𝑥𝑦 and 𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremvpwex 5089 Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 5090 from vpwex 5089. (Revised by BJ, 10-Aug-2022.)
𝒫 𝑥 ∈ V

Theorempwexg 5090 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
(𝐴𝑉 → 𝒫 𝐴 ∈ V)

Theorempwexd 5091 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)

Theorempwex 5092 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       𝒫 𝐴 ∈ V

Theoremabssexg 5093* 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 5094 Alternate proof of snex 5140 using Power Set (ax-pow 5077) instead of Pairing (ax-pr 5138). Unlike in the proof of zfpair 5136, Replacement (ax-rep 5006) 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 5095 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5096. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V

Theoremp0exALT 5096 Alternate proof of p0ex 5095 which is quite different and longer if snexALT 5094 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
{∅} ∈ V

Theorempp0ex 5097 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
{∅, {∅}} ∈ V

Theoremord3ex 5098 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7226. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} ∈ V

TheoremdtruALT 5099* Alternate proof of dtru 5082 which requires more axioms but is shorter and may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3502 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the \$d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

¬ ∀𝑥 𝑥 = 𝑦

Theoremaxc16b 5100* This theorem shows that axiom ax-c16 35046 is redundant in the presence of theorem dtru 5082, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

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