P. 377 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-tagci 37601	Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ tag 𝐵)
Theorem	bj-tagcg 37602	Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Theorem	bj-taginv 37603*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Theorem	bj-tagex 37604	A class is a set if and only if its tagging is a set. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V)
Theorem	bj-xtageq 37605	The products of a given class and the tagging of either of two equal classes are equal. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 × tag 𝐴) = (𝐶 × tag 𝐵))
Theorem	bj-xtagex 37606	The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V))
21.19.5.15  Tuples of classes This subsection gives a definition of an ordered pair, or couple (2-tuple), that "works" for proper classes, as evidenced by Theorems bj-2uplth 37638 and bj-2uplex 37639, and more importantly, bj-pr21val 37630 and bj-pr22val 37636. In particular, one can define well-behaved tuples of classes. Classes in ZF(C) are only virtual, and in particular they cannot be quantified over. Theorem bj-2uplex 37639 has advantages: in view of df-br 5111, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5792 (resp. relsnop 5794) would hold without antecedents (resp. hypotheses) thanks to relsnb 5791). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5715 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 17208 could be simplified by removing the exception currently made for the empty set. The projections are denoted by pr1 and pr2 and the couple with projections (or coordinates) 𝐴 and 𝐵 is denoted by ⦅𝐴, 𝐵⦆. Note that this definition uses the Kuratowski definition (df-op 4597) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9588) without needing the axiom of regularity; it could even bypass this definition by "inlining" it. This definition is due to Anthony Morse and is expounded (with idiosyncratic notation) in Anthony P. Morse, A Theory of Sets, Academic Press, 1965 (second edition 1986). Note that this extends in a natural way to tuples. A variation of this definition is justified in opthprc 5727, but here we use "tagged versions" of the factors (see df-bj-tag 37592) so that an m-tuple can equal an n-tuple only when m = n (and the projections are the same). A comparison of the different definitions of tuples (strangely not mentioning Morse's), is given in Dominic McCarty and Dana Scott, Reconsidering ordered pairs, Bull. Symbolic Logic, Volume 14, Issue 3 (Sept. 2008), 379--397. where a recursive definition of tuples is given that avoids the two-step definition of tuples and that can be adapted to various set theories. Finally, another survey is Akihiro Kanamori, The empty set, the singleton, and the ordered pair, Bull. Symbolic Logic, Volume 9, Number 3 (Sept. 2003), 273--298. (available at http://math.bu.edu/people/aki/8.pdf 37592)
Syntax	bj-cproj 37607	Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
Definition	df-bj-proj 37608*	Definition of the class projection corresponding to tagged tuples. The expression (𝐴 Proj 𝐵) denotes the projection on the A^th component. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
Theorem	bj-projeq 37609	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Theorem	bj-projeq2 37610	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
Theorem	bj-projun 37611	The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
Theorem	bj-projex 37612	Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)
Theorem	bj-projval 37613	Value of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))
Syntax	bj-c1upl 37614	Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class ⦅𝐴⦆
Definition	df-bj-1upl 37615	Definition of the Morse monuple (1-tuple). This is not useful per se, but is used as a step towards the definition of couples (2-tuples, or ordered pairs). The reason for "tagging" the set is so that an m-tuple and an n-tuple be equal only when m = n. Note that with this definition, the 0-tuple is the empty set. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 37629, bj-2uplth 37638, bj-2uplex 37639, and the properties of the projections (see df-bj-pr1 37618 and df-bj-pr2 37632). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
Theorem	bj-1upleq 37616	Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
Syntax	bj-cpr1 37617	Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
Definition	df-bj-pr1 37618	Definition of the first projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr1eq 37619, bj-pr11val 37622, bj-pr21val 37630, bj-pr1ex 37623. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ pr1 𝐴 = (∅ Proj 𝐴)
Theorem	bj-pr1eq 37619	Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Theorem	bj-pr1un 37620	The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Theorem	bj-pr1val 37621	Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Theorem	bj-pr11val 37622	Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ⦅𝐴⦆ = 𝐴
Theorem	bj-pr1ex 37623	Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V)
Theorem	bj-1uplth 37624	The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
Theorem	bj-1uplex 37625	A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
Theorem	bj-1upln0 37626	A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ ⦅𝐴⦆ ≠ ∅
Syntax	bj-c2uple 37627	Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class ⦅𝐴, 𝐵⦆
Definition	df-bj-2upl 37628	Definition of the Morse couple. See df-bj-1upl 37615. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 37629, bj-2uplth 37638, bj-2uplex 37639, and the properties of the projections (see df-bj-pr1 37618 and df-bj-pr2 37632). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
Theorem	bj-2upleq 37629	Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Theorem	bj-pr21val 37630	Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
Syntax	bj-cpr2 37631	Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
Definition	df-bj-pr2 37632	Definition of the second projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr2eq 37633, bj-pr22val 37636, bj-pr2ex 37637. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ pr2 𝐴 = (1o Proj 𝐴)
Theorem	bj-pr2eq 37633	Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Theorem	bj-pr2un 37634	The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Theorem	bj-pr2val 37635	Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Theorem	bj-pr22val 37636	Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
Theorem	bj-pr2ex 37637	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 37638	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5460). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 37639	A couple is a set if and only if its coordinates are sets. For the advantages offered by the reverse closure property, see the section head comment. (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-2upln0 37640	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 37641	A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have ⦅𝐴, ∅⦆ = ⦅𝐴⦆. Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 37626 and bj-2upln0 37640 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ⦅𝐶⦆
21.19.5.16  Set theory: elementary operations relative to a universe Some elementary set-theoretic operations "relative to a universe" (by which is merely meant some given class considered as a universe).
Theorem	bj-rcleqf 37642	Relative version of cleqf 2953. (Contributed by BJ, 27-Dec-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑉    ⇒   ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-rcleq 37643*	Relative version of dfcleq 2756. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-reabeq 37644*	Relative form of eqabb 2902. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	bj-disj2r 37645	Relative version of ssdifin0 4447, allowing a biconditional, and of disj2 4419. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4447 nor disj2 4419. (Proof modification is discouraged.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
Theorem	bj-sscon 37646	Contraposition law for relative subclasses. Relative and generalized version of ssconb 4097. Shortens ssconb 4097, conss2 45135. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4097 nor conss2 45135. (Proof modification is discouraged.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴))
21.19.5.17  Axioms for finite unions In this section, we introduce the axiom of singleton ax-bj-sn 37650 and the axiom of binary union ax-bj-bun 37654. Both axioms are implied by the standard axioms of unordered pair ax-pr 5406 and of union ax-un 7734 (see snex 5412 and unex 7744). Conversely, the axiom of unordered pair ax-pr 5406 is implied by the axioms of singleton and of binary union, as proved in bj-prexg 37656 and bj-prex 37657. The axioms of union ax-un 7734 and of powerset ax-pow 5338 are independent of these axioms: consider respectively the class of pseudo-hereditarily sets of cardinality less than a given singular strong limit cardinal, see Greg Oman, On the axiom of union, Arch. Math. Logic (2010) 49:283--289 (that model does have finite unions), and the class of well-founded hereditarily countable sets (or hereditarily less than a given uncountable regular cardinal). See also https://mathoverflow.net/questions/81815 5338 and https://mathoverflow.net/questions/48365 5338. A proof by finite induction shows that the existence of finite unions is equivalent to the existence of binary unions and of nullary unions (the latter being the axiom of the empty set ax-nul 5270). The axiom of binary union is useful in theories without the axioms of union ax-un 7734 and of powerset ax-pow 5338. For instance, the class of well-founded sets hereditarily of cardinality at most 𝑛 ∈ ℕ0 with ordinary membership relation is a model of { ax-ext 2735, ax-rep 5239, ax-sep 5258, ax-nul 5270, ax-reg 9555 } and the axioms of existence of unordered 𝑚-tuples for all 𝑚 ≤ 𝑛, and in most cases one would like to rule out such models, hence the need for extra axioms, typically variants of powersets or unions. The axiom of adjunction ax-bj-adj 37659 is more widely used, and is an axiom of General Set Theory. We prove how to retrieve it from binary union and singleton in bj-adjfrombun 37663 and conversely how to prove from adjunction singleton (bj-snfromadj 37661) and unordered pair (bj-prfromadj 37662).
Theorem	bj-abex 37647*	Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-clex 37648*	Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   ⊢ (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-axsn 37649*	Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37650). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
Axiom	ax-bj-sn 37650*	Axiom of singleton. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
Theorem	bj-snexg 37651	A singleton built on a set is a set. Contrary to bj-snex 37652, this proof is intuitionistically valid and does not require ax-nul 5270. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5412 and prove it from ax-bj-sn 37650. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 37652	A singleton is a set. See also snex 5412, snexALT 5356. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37650. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴} ∈ V
Theorem	bj-axbun 37653*	Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 37654). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))
Axiom	ax-bj-bun 37654*	Axiom of binary union. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
Theorem	bj-unexg 37655	Existence of binary unions of sets, proved from ax-bj-bun 37654. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-prexg 37656	Existence of unordered pairs formed on sets, proved from ax-bj-sn 37650 and ax-bj-bun 37654. Contrary to bj-prex 37657, this proof is intuitionistically valid and does not require ax-nul 5270. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-prex 37657	Existence of unordered pairs proved from ax-bj-sn 37650 and ax-bj-bun 37654. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	bj-axadj 37658*	Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37659). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))
Axiom	ax-bj-adj 37659*	Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
Theorem	bj-adjg1 37660	Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)
Theorem	bj-snfromadj 37661	Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥} ∈ V
Theorem	bj-prfromadj 37662	Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-adjfrombun 37663	Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∪ {𝑦}) ∈ V
21.19.5.18  Set theory: miscellaneous Miscellaneous theorems of set theory.
Theorem	eleq2w2ALT 37664	Alternate proof of eleq2w2 2759 and special instance of eleq2 2852. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-clel3gALT 37665*	Alternate proof of clel3g 3621. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
Theorem	bj-pw0ALT 37666	Alternate proof of pw0 4779. The proofs have a similar structure: pw0 4779 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 37666 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4779 and biconditional for bj-pw0ALT 37666) to translate the property ss0b 4359 into the wanted result. To translate a biconditional into a class equality, pw0 4779 uses abbii 2830 (which yields an equality of class abstractions), while bj-pw0ALT 37666 uses eqriv 2760 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2830, through its closed form abbi 2828, is proved from eqrdv 2761, which is the deduction form of eqriv 2760. In the other direction, velpw 4568 and velsn 4606 are proved from the definitions of powerclass and singleton using elabg 3636, which is a version of abbii 2830 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝒫 ∅ = {∅}
Theorem	bj-sselpwuni 37667	Quantitative version of ssexg 5291: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)
Theorem	bj-unirel 37668	Quantitative version of uniexr 7763: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)
⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
Theorem	bj-elpwg 37669	If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4566 and elpw2g 5305 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	bj-velpwALT 37670*	This theorem bj-velpwALT 37670 and the next theorem bj-elpwgALT 37671 are alternate proofs of velpw 4568 and elpwg 4566 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3525 instead of proving first the general case using elab2g 3640 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2213. In other cases, that order is better (e.g., vsnex 5408 proved before snexg 5413). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
Theorem	bj-elpwgALT 37671	Alternate proof of elpwg 4566. See comment for bj-velpwALT 37670. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	bj-vjust 37672	Justification theorem for dfv2 3458 if it were the definition. See also vjust 3456. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Theorem	bj-nul 37673*	Two formulations of the axiom of the empty set ax-nul 5270. Proposal: place it right before ax-nul 5270. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliota 37674*	Definition of the empty set using the definite description binder. See also bj-nuliotaALT 37675. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliotaALT 37675*	Alternate proof of bj-nuliota 37674. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6518). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-vtoclgfALT 37676	Alternate proof of vtoclgf 3535. Proof from vtoclgft 3521. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-elsn12g 37677	Join of elsng 4604 and elsn2g 4631. (Contributed by BJ, 18-Nov-2023.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-elsnb 37678	Biconditional version of elsng 4604. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-pwcfsdom 37679	Remove hypothesis from pwcfsdom 10569. Illustration of how to remove a "proof-facilitating hypothesis". Shortens theorems using pwcfsdom 10569. (Contributed by BJ, 14-Sep-2019.)
⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
Theorem	bj-grur1 37680	Remove hypothesis from grur1 10806. Illustration of how to remove a "definitional hypothesis". This makes its uses longer, but the theorem feels more self-contained. It looks preferable when the defined term appears only once in the conclusion. (Contributed by BJ, 14-Sep-2019.)
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘(𝑈 ∩ On)))
Theorem	bj-bm1.3ii 37681*	The extension of a predicate (𝜑(𝑧)) is included in a set (𝑥) if and only if it is a set (𝑦). Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5258. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.) Generalized to a closed form biconditional with existential quantifications using two different setvars 𝑥, 𝑦 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.) TODO: move after sepexi 5265. Relabel ("sepbi"?).
⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-dfid2ALT 37682	Alternate version of dfid2 5560. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5558 instead to make the semantics of the construction df-opab 5175 clearer. (New usage is discouraged.)
⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
Theorem	bj-0nelopab 37683	The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by BJ, 22-Jul-2023.) TODO: move to the main section when one can reorder sections so that we can use relopab 5813 (this is a very limited reordering).
⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	bj-brrelex12ALT 37684	Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5715. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-epelg 37685	The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5566 and closed form of epeli 5565. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5719 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	bj-epelb 37686	Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5720 available. Check if it is shorter to prove bj-epelg 37685 first or bj-epelb 37686 first. (Contributed by BJ, 14-Jul-2023.)
⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
Theorem	bj-nsnid 37687	A set does not contain the singleton formed on it. More precisely, one can prove that a class contains the singleton formed on it if and only if it is proper and contains the empty set (since it is "the singleton formed on" any proper class, see snprc 4684): ⊢ ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴 → 𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴)
Theorem	bj-rdg0gALT 37688	Alternate proof of rdg0g 8415. More direct since it bypasses tz7.44-1 8394 and rdg0 8409 (and vtoclg 3523, vtoclga 3542). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	bj-vn0ALT 37689	Alternate proof of vn0 4299 which does not use eqabbw 2836 (and is shorter than vn0 4299 when eqabbw 2836 is inlined). (Contributed by BJ, 12-Jul-2026.) Using the same dummy variable for 𝑦 and 𝑧 slightly reduces the proof size. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ V ≠ ∅
21.19.5.19  Axioms of separation and replacement This section proves basic relations among some standard axioms of set theory, in particular the axiom of separation (the universal closure of ax-sep 5258) and the version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function, bj-rep 37691. These axioms often appear (as specific instances) in the hypotheses of the theorems in this section.
Theorem	bj-axnul 37690*	Over the base theory ax-1 6-- ax-5 1940, the axiom of separation implies the weak emptyset axiom. By "weak emptyset axiom", we mean the axiom asserting existence of an empty set (which can be called "the" empty set when the axiom of extensionality ax-ext 2735 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 2005). This is the conclusion of bj-axnul 37690. Note that the weak emptyset axiom implies ⊢ (∃𝑥⊤ → ∃𝑦⊤) without DV conditions hence also the same statement as the weak emptyset axiom without DV conditions on 𝑥, but only on 𝑦, 𝑧. By "axiom of separation", we mean the universal closure of ax-sep 5258, simulated here by its instance with ⊥ substituted for 𝜑 (and with the variable used to assert existence in the weak emptyset axiom substituted for the containing set) as the hypothesis of bj-axnul 37690. In particular, the axiom of existence extru 2005 and the axiom of separation together imply the emptyset axiom (and conversely, the emptyset axiom implies the axiom of existence). Note: this theorem does not require a disjointness condition on 𝑦, 𝑧, although both axioms should be stated with all variables disjoint. This proof only uses an instance of the axiom of separation with a bounded formula, so is valid in a constructive setting (see the CZF section in the "Intuitionistic Logic Explorer" iset.mm). (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥))    ⇒   ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
Theorem	bj-rep 37691*	Version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function from ax-rep 5239 (in the form of axrep6 5248). (Contributed by BJ, 14-Mar-2026.) The proof proves the statement without the DV condition on 𝑥, 𝜑, but the DV condition is added to this statement to show that this weaker version is sufficient. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
Theorem	bj-axseprep 37692*	Axiom of separation (universal closure of ax-sep 5258) from a weak form of the axiom of replacement requiring that the functional relation in it be a (total) function and the weak emptyset axiom (existence of an empty set provided existence of a set), as written in the theorem's hypotheses. This result shows that the weak emptyset axiom is not only the result of a cheap way to avoid an axiom redundancy (in this case, the existence axiom extru 2005) by adding it as an antecedent, but also permits to prove nontrivial results that hold in nonnecessarily nonempty universes. This proof is by cases so is not intuitionistic. The statement does not require a nonempty universe; most of the proof does not either, and the parts that do (e.g., near sb8ef 2387 and sbequ12r 2288 and eueq2 3674) could be reworked to avoid it. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1957. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)    &   ⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓))    &   ⊢ (𝜓 ↔ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))    ⇒   ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))
Theorem	bj-axreprepsep 37693*	Strong axiom of replacement (universal closure of ax-rep 5239) from the axioms of separation and replacement as written in the theorem's hypotheses. The statement does not require a nonempty universe; most of the proof does not either, except for the use of 19.8a 2217, which could be removed by reworking the proof, since it is applied in a subexpression bound by the variable it introduces. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1957. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑠∀𝑦(𝑦 ∈ 𝑠 ↔ (𝑦 ∈ 𝑥 ∧ ∃𝑧𝜑))    &   ⊢ ∀𝑠(∀𝑦 ∈ 𝑠 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑠 𝜑))    ⇒   ⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
21.19.5.20  Evaluation at a class This section treats the existing predicate Slot (df-slot 17243) as "evaluation at a class" and for the moment does not introduce new syntax for it.
Theorem	bj-evaleq 37694	Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17244 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Theorem	bj-evalfun 37695	The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
⊢ Fun Slot 𝐴
Theorem	bj-evalfn 37696	The evaluation at a class is a function on the universal class. (General form of slotfn 17245). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
⊢ Slot 𝐴 Fn V
Theorem	bj-evalf 37697	The evaluation at a class is a function from the universal class into the universal class. (Contributed by BJ, 17-Mar-2026.)
⊢ Slot 𝐴:V⟶V
Theorem	bj-evalval 37698	Value of the evaluation at a class. Closed form of strfvnd 17246 and strfvn 17247. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))
Theorem	bj-evalid 37699	The evaluation at a set of the identity function is that set. General form of ndxarg 17257. The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
Theorem	bj-ndxarg 37700	Proof of ndxarg 17257 from bj-evalid 37699. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁