P. 54 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	intexab 5301	The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrab 5302	The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	iinexg 5303*	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 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	intabs 5304*	Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝜑 ↔ 𝜒))    &   ⊢ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)    ⇒   ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}
Theorem	inuni 5305*	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.) (Proof shortened by Wolf Lammen, 15-May-2025.)
⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}
Theorem	axpweq 5306*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5320 is not used by the proof. When ax-pow 5320 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 7741). (Contributed by NM, 22-Jun-2009.)
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
Theorem	pwnss 5307	The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof shortened by BJ, 24-Jul-2025.)
⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
Theorem	pwne 5308	No set equals its power set. The sethood antecedent is necessary; compare pwv 4868. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	difelpw 5309	A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)
2.2.5  Theorems requiring empty set existence
Theorem	class2set 5310*	The class of elements of 𝐴 "such that 𝐴 is a set" is a set. That class is equal to 𝐴 when 𝐴 is a set (see class2seteq 3675) and to the empty set when 𝐴 is a proper class. (Contributed by NM, 16-Oct-2003.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V
Theorem	0elpw 5311	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	pwne0 5312	A power class is never empty. (Contributed by NM, 3-Sep-2018.)
⊢ 𝒫 𝐴 ≠ ∅
Theorem	0nep0 5313	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 5314	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 = {∅})
Theorem	unidif0 5315	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	eqsnuniex 5316	If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.)
⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
Theorem	iin0 5317*	An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
Theorem	notzfaus 5318*	In the Separation Scheme zfauscl 5253, 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.) (Proof shortened by BJ, 18-Nov-2023.)
⊢ 𝐴 = {∅}    &   ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)    ⇒   ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	intv 5319	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets
Axiom	ax-pow 5320*	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 5322 uses explicit subset notation. A version using class notation is pwex 5335. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 5321*	Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpow2 5322*	A variant of the Axiom of Power Sets ax-pow 5320 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 5323*	A variant of the Axiom of Power Sets ax-pow 5320. 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.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)
Theorem	elALT2 5324*	Alternate proof of el 5397 using ax-9 2119 and ax-pow 5320 instead of ax-pr 5387. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	dtruALT2 5325*	Alternate proof of dtru 5396 using ax-pow 5320 instead of ax-pr 5387. See dtruALT 5343 for another proof using ax-pow 5320 instead of ax-pr 5387. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-12 2178. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	dtrucor 5326*	Corollary of dtru 5396. 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 5327. (Contributed by NM, 27-Jun-2002.)
⊢ 𝑥 = 𝑦    ⇒   ⊢ 𝑥 ≠ 𝑦
Theorem	dtrucor2 5327	The theorem form of the deduction dtrucor 5326 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5326. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
Theorem	dvdemo1 5328*	Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑦 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑧). That theorem bundles the theorems (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) with 𝑥, 𝑦 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥) with 𝑥, 𝑦 disjoint). Compare with dvdemo2 5329, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance. See https://us.metamath.org/mpeuni/mmset.html#distinct 5329 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5328 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2158 nor ax-13 2370. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	dvdemo2 5329*	Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑧 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑦). That theorem bundles the theorems (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑧 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint). Compare with dvdemo1 5328, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance. See https://us.metamath.org/mpeuni/mmset.html#distinct 5328 for details on the "disjoint variable" mechanism. Note that dvdemo2 5329 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑦 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 2142, ax-11 2158, ax-12 2178, ax-13 2370. (Contributed by NM, 1-Dec-2006.) Avoid ax-13 2370. (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	nfnid 5330	A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtruALT2 5325 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ ¬ Ⅎ𝑥𝑥
Theorem	nfcvb 5331	The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of Ⅎ in the obvious way. This theorem is not true in a one-element domain, because then Ⅎ𝑥𝑦 and ∀𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2370. (New usage is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	vpwex 5332	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 5333 from vpwex 5332. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 5333	Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
Theorem	pwexd 5334	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	pwex 5335	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	pwel 5336	Quantitative version of pwexg 5333: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5261 and ax-pr 5387 and shorten proof. (Revised by BJ, 13-Apr-2024.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	abssexg 5337*	Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
Theorem	snexALT 5338	Alternate proof of snex 5391 using Power Set (ax-pow 5320) instead of Pairing (ax-pr 5387). Unlike in the proof of zfpair 5376, Replacement (ax-rep 5234) 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
Theorem	p0ex 5339	The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5340. (Contributed by NM, 23-Dec-1993.)
⊢ {∅} ∈ V
Theorem	p0exALT 5340	Alternate proof of p0ex 5339 which is quite different and longer if snexALT 5338 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {∅} ∈ V
Theorem	pp0ex 5341	The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
⊢ {∅, {∅}} ∈ V
Theorem	ord3ex 5342	The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7711. (Contributed by NM, 2-May-2009.)
⊢ {∅, {∅}, {∅, {∅}}} ∈ V
Theorem	dtruALT 5343*	Alternate proof of dtru 5396 which requires more axioms but is shorter and may be easier to understand. Like dtruALT2 5325, it uses ax-pow 5320 rather than ax-pr 5387. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, Theorem spcev 3572 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
Theorem	axc16b 5344*	This theorem shows that Axiom ax-c16 38885 is redundant in the presence of Theorem dtruALT2 5325, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5325 (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	eunex 5345	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.) (Proof shortened by BJ, 2-Jan-2023.)
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Theorem	eusv1 5346*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)
Theorem	eusvnf 5347*	Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
Theorem	eusvnfb 5348*	Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
Theorem	eusv2i 5349*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
Theorem	eusv2nf 5350*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
Theorem	eusv2 5351*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴)
Theorem	reusv1 5352*	Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
Theorem	reusv2lem1 5353*	Lemma for reusv2 5358. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	reusv2lem2 5354*	Lemma for reusv2 5358. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
Theorem	reusv2lem3 5355*	Lemma for reusv2 5358. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	reusv2lem4 5356*	Lemma for reusv2 5358. (Contributed by NM, 13-Dec-2012.)
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
Theorem	reusv2lem5 5357*	Lemma for reusv2 5358. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))
Theorem	reusv2 5358*	Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦 ∈ 𝐵 such that 𝜑. The first antecedent ensures that the constant value belongs to the existential uniqueness domain 𝐴, and the second ensures that 𝐶(𝑦) is evaluated for at least one 𝑦. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
Theorem	reusv3i 5359*	Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
Theorem	reusv3 5360*	Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5352 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)    ⇒   ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
Theorem	eusv4 5361*	Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
Theorem	alxfr 5362*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	ralxfrd 5363*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))
Theorem	rexxfrd 5364*	Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
Theorem	ralxfr2d 5365*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))
Theorem	rexxfr2d 5366*	Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
Theorem	ralxfrd2 5367*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5363. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))
Theorem	rexxfrd2 5368*	Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5364. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
Theorem	ralxfr 5369*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)
Theorem	ralxfrALT 5370*	Alternate proof of ralxfr 5369 which does not use ralxfrd 5363. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)
Theorem	rexxfr 5371*	Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓)
Theorem	rabxfrd 5372*	Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 16-Jan-2012.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
Theorem	rabxfr 5373*	Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 10-Jun-2005.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
Theorem	reuhypd 5374*	A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7378. (Contributed by NM, 16-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	reuhyp 5375*	A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3723. (Contributed by NM, 15-Nov-2004.)
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)    &   ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	zfpair 5376	The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms. This theorem should not be referenced by any proof other than axprALT 5377. Instead, use zfpair2 5388 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	axprALT 5377*	Alternate proof of axpr 5382. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
2.3.2  Derive the Axiom of Pairing
Theorem	axprlem1 5378*	Lemma for axpr 5382. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)
Theorem	axprlem2 5379*	Lemma for axpr 5382. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)
Theorem	axprlem3 5380*	Lemma for axpr 5382. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by Matthew House, 18-Sep-2025.)
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem4 5381*	Lemma for axpr 5382. If an existing set of empty sets corresponds to one element of the pair, then the element is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) (Revised by Matthew House, 18-Sep-2025.)
⊢ ∃𝑠∀𝑛𝜑    &   ⊢ (𝜑 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))    &   ⊢ (∀𝑛𝜑 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑣))    ⇒   ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (𝑤 = 𝑣 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
Theorem	axpr 5382*	Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms. This theorem should not be referenced by any proof. Instead, use ax-pr 5387 below so that the uses of the Axiom of Pairing can be more easily identified. For a shorter proof using ax-ext 2701, see axprALT 5377. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2701. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) (Proof shortened by Matthew House, 18-Sep-2025.) Use ax-pr 5387 instead. (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	axprlem3OLD 5383*	Obsolete version of axprlem3 5380 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem4OLD 5384*	Obsolete version of axprlem4 5381 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem5OLD 5385*	Obsolete version of axprlem4 5381 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprOLD 5386*	Obsolete version of axpr 5382 as of 18-Sep-2025. (Contributed by NM, 14-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Axiom	ax-pr 5387*	The Axiom of Pairing of ZF set theory. It was derived as Theorem axpr 5382 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, 14-Nov-2006.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	zfpair2 5388	Derive the abbreviated version of the Axiom of Pairing from ax-pr 5387. See zfpair 5376 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	vsnex 5389	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
⊢ {𝑥} ∈ V
Theorem	snexg 5390	A singleton built on a set is a set. Special case of snex 5391 which does not require ax-nul 5261 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5391 and shorten proof. (Revised by BJ, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	snex 5391	A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 5338. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.)
⊢ {𝐴} ∈ V
Theorem	prex 5392	The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 4731), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	exel 5393*	There exist two sets, one a member of the other. This theorem looks similar to el 5397, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1809, ax-6 1967, and ax-pr 5387. This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq 5394. (Contributed by SN, 23-Dec-2024.)
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦
Theorem	exexneq 5394*	There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2111. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2178. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5387 instead of ax-pow 5320. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5396. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
Theorem	exneq 5395*	Given any set (the "𝑦 " in the statement), there exists a set not equal to it. The same statement without disjoint variable condition is false, since we do not have ∃𝑥¬ 𝑥 = 𝑥. This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext 2701, ax-sep 5251, or ax-pow 5320 nor auxiliary logical axiom schemes ax-10 2142 to ax-13 2370. See dtruALT 5343 for a shorter proof using more axioms, and dtruALT2 5325 for a proof using ax-pow 5320 instead of ax-pr 5387. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2111. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2178. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5387 instead of ax-pow 5320. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5396. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥 ¬ 𝑥 = 𝑦
Theorem	dtru 5396*	Given any set (the "𝑦 " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 2028. The same comments and revision history concerning axiom usage as in exneq 5395 apply. See dtruALT 5343 and dtruALT2 5325 for alternate proofs avoiding ax-pr 5387. (Contributed by NM, 7-Nov-2006.) Extract exneq 5395 as an intermediate result. (Revised by BJ, 2-Jan-2025.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	el 5397*	Any set is an element of some other set. See elALT 5400 for a shorter proof using more axioms, and see elALT2 5324 for a proof that uses ax-9 2119 and ax-pow 5320 instead of ax-pr 5387. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5387 instead of ax-9 2119 and ax-pow 5320. (Revised by BTernaryTau, 2-Dec-2024.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	sels 5398*	If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5400. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5251, ax-nul 5261, ax-pow 5320. (Revised by BTernaryTau, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	selsALT 5399*	Alternate proof of sels 5398, requiring ax-sep 5251 but not using el 5397 (which is proved from it as elALT 5400). (especially when the proof of el 5397 is inlined in sels 5398). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5400. (Revised by BJ, 3-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	elALT 5400*	Alternate proof of el 5397, shorter but requiring ax-sep 5251. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦