P. 336 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33501-33600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-rabtrALT 33501*	Alternate proof of bj-rabtr 33500. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrAUTO 33502*	Proof of bj-rabtr 33500 found automatically by "MM-PA> IMPROVE ALL / DEPTH 3 / 3" followed by "MM-PA> MINIMIZEWITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
20.14.5.9  Restricted non-freeness In this subsection, we define restricted non-freeness (or relative non-freeness).
Syntax	wrnf 33503	Syntax for restricted non-freeness.
wff Ⅎ𝑥 ∈ 𝐴𝜑
Definition	df-bj-rnf 33504	Definition of restricted non-freeness. Informally, the proposition Ⅎ𝑥 ∈ 𝐴𝜑 means that 𝜑(𝑥) does not vary on 𝐴. (Contributed by BJ, 19-Mar-2021.)
⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
20.14.5.10  Russell's paradox A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 33505) and then two versions (bj-ru1 33506 and bj-ru 33507). Special attention is put on minimizing axiom depencencies.
Theorem	bj-ru0 33505*	The FOL part of Russell's paradox ru 3650 (see also bj-ru1 33506, bj-ru 33507). Use of elequ1 2113, bj-elequ12 33257, bj-spvv 33311 (instead of eleq1 2846, eleq12d 2852, spv 2357 as in ru 3650) permits to remove dependency on ax-10 2134, ax-11 2149, ax-12 2162, ax-13 2333, ax-ext 2753, df-sb 2012, df-clab 2763, df-cleq 2769, df-clel 2773. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
Theorem	bj-ru1 33506*	A version of Russell's paradox ru 3650 (see also bj-ru 33507). Note the more economical use of bj-abeq2 33350 instead of abeq2 2891 to avoid dependency on ax-13 2333. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
Theorem	bj-ru 33507	Remove dependency on ax-13 2333 (and df-v 3399) from Russell's paradox ru 3650 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 5018 does require ax-8 2108 and ax-9 2115 since it requires df-clel 2773 and df-cleq 2769--- see bj-df-clel 33459 and bj-df-cleq 33464). Note the more economical use of bj-elissetv 33431 instead of isset 3408 to avoid use of df-v 3399. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉
20.14.5.11  Some disjointness results A few utility theorems on disjointness of classes.
Theorem	bj-n0i 33508*	Inference associated with n0 4158. Shortens 2ndcdisj 21668 (2888>2878), notzfaus 5074 (264>253). (Contributed by BJ, 22-Apr-2019.)
⊢ 𝐴 ≠ ∅    ⇒   ⊢ ∃𝑥 𝑥 ∈ 𝐴
Theorem	bj-disjcsn 33509	A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 31405 and does not depend on df-ne 2969. (Contributed by BJ, 4-Apr-2019.)
⊢ (𝐴 ∩ {𝐴}) = ∅
Theorem	bj-disjsn01 33510	Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33509 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
⊢ ({∅} ∩ {1o}) = ∅
Theorem	bj-2ex 33511	2o is a set. (Contributed by BJ, 6-Apr-2019.)
⊢ 2o ∈ V
Theorem	bj-0nel1 33512	The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∉ {1o}
Theorem	bj-1nel0 33513	1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
⊢ 1o ∉ {∅}
20.14.5.12  Complements on direct products A few utility theorems on direct products.
Theorem	bj-xpimasn 33514	The image of a singleton, general case. [Change and relabel xpimasn 5833 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
Theorem	bj-xpima1sn 33515	The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5833 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ (𝑋 ∉ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima1snALT 33516	Alternate proof of bj-xpima1sn 33515. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑋 ∉ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima2sn 33517	The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5833] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	bj-xpnzex 33518	If the first factor of a product is nonempty, and the product is a set, then the second factor is a set. UPDATE: this is actually the curried (exported) form of xpexcnv 7387 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉 → 𝐵 ∈ V))
Theorem	bj-xpexg2 33519	Curried (exported) form of xpexg 7237. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V))
Theorem	bj-xpnzexb 33520	If the first factor of a product is a nonempty set, then the product is a set if and only if the second factor is a set. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))
Theorem	bj-cleq 33521*	Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
20.14.5.13  "Singletonization" and tagging This subsection introduces the "singletonization" and the "tagging" of a class. The singletonization of a class is the class of singletons of elements of that class. It is useful since all nonsingletons are disjoint from it, so one can easily adjoin to it disjoint elements, which is what the tagging does: it adjoins the empty set. This can be used for instance to define the one-point compactification of a topological space. It will be used in the next section to define tuples which work for proper classes.
Theorem	bj-sels 33522*	If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	bj-snsetex 33523*	The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5006. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Theorem	bj-clex 33524*	Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
Syntax	bj-csngl 33525	Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
Definition	df-bj-sngl 33526*	Definition of "singletonization". The class sngl 𝐴 is isomorphic to 𝐴 and since it contains only singletons, it can be easily be adjoined disjoint elements, which can be useful in various constructions. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
Theorem	bj-sngleq 33527	Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Theorem	bj-elsngl 33528*	Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
Theorem	bj-snglc 33529	Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)
Theorem	bj-snglss 33530	The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-0nelsngl 33531	The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7843). (Contributed by BJ, 6-Oct-2018.)
⊢ ∅ ∉ sngl 𝐴
Theorem	bj-snglinv 33532*	Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
Theorem	bj-snglex 33533	A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Syntax	bj-ctag 33534	Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
Definition	df-bj-tag 33535	Definition of the tagged copy of a class, that is, the adjunction to (an isomorph of) 𝐴 of a disjoint element (here, the empty set). Remark: this could be used for the one-point compactification of a topological space. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
Theorem	bj-tageq 33536	Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
Theorem	bj-eltag 33537*	Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Theorem	bj-0eltag 33538	The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∈ tag 𝐴
Theorem	bj-tagn0 33539	The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ tag 𝐴 ≠ ∅
Theorem	bj-tagss 33540	The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-snglsstag 33541	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ tag 𝐴
Theorem	bj-sngltagi 33542	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵)
Theorem	bj-sngltag 33543	The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Theorem	bj-tagci 33544	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 33545	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 33546*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Theorem	bj-tagex 33547	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 33548	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 33549	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))
20.14.5.14  Tuples of classes This subsection gives a definition of an ordered pair, or couple (2-tuple), which "works" for proper classes, as evidenced by Theorems bj-2uplth 33581 and bj-2uplex 33582 (but more importantly, bj-pr21val 33573 and bj-pr22val 33579). 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 33582 has advantages: in view of df-br 4887, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5474 (resp. relsnop 5476) would hold without antecedents (resp. hypotheses) thanks to relsnb 5473). Similarly, df-struct 16257 could be simplified with the exception of the empty set removed. 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 Kuratowksi definition (df-op 4404) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 8810) 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 5413, but here we use "tagged versions" of the factors (see df-bj-tag 33535) 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 2-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)
Syntax	bj-cproj 33550	Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
Definition	df-bj-proj 33551*	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 33552	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Theorem	bj-projeq2 33553	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
Theorem	bj-projun 33554	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 33555	Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)
Theorem	bj-projval 33556	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 33557	Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class ⦅𝐴⦆
Definition	df-bj-1upl 33558	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 33572, bj-2uplth 33581, bj-2uplex 33582, and the properties of the projections (see df-bj-pr1 33561 and df-bj-pr2 33575). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
Theorem	bj-1upleq 33559	Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
Syntax	bj-cpr1 33560	Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
Definition	df-bj-pr1 33561	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 33562, bj-pr11val 33565, bj-pr21val 33573, bj-pr1ex 33566. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ pr1 𝐴 = (∅ Proj 𝐴)
Theorem	bj-pr1eq 33562	Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Theorem	bj-pr1un 33563	The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Theorem	bj-pr1val 33564	Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Theorem	bj-pr11val 33565	Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ⦅𝐴⦆ = 𝐴
Theorem	bj-pr1ex 33566	Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V)
Theorem	bj-1uplth 33567	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 33568	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 33569	A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ ⦅𝐴⦆ ≠ ∅
Syntax	bj-c2uple 33570	Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class ⦅𝐴, 𝐵⦆
Definition	df-bj-2upl 33571	Definition of the Morse couple. See df-bj-1upl 33558. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 33572, bj-2uplth 33581, bj-2uplex 33582, and the properties of the projections (see df-bj-pr1 33561 and df-bj-pr2 33575). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
Theorem	bj-2upleq 33572	Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Theorem	bj-pr21val 33573	Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
Syntax	bj-cpr2 33574	Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
Definition	df-bj-pr2 33575	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 33576, bj-pr22val 33579, bj-pr2ex 33580. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ pr2 𝐴 = (1o Proj 𝐴)
Theorem	bj-pr2eq 33576	Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Theorem	bj-pr2un 33577	The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Theorem	bj-pr2val 33578	Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Theorem	bj-pr22val 33579	Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
Theorem	bj-pr2ex 33580	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 33581	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5176). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 33582	A couple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-2upln0 33583	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 33584	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 33569 and bj-2upln0 33583 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⦆
20.14.5.15  Set theory: miscellaneous Miscellaneous theorems of set theory.
Theorem	bj-disj2r 33585	Relative version of ssdifin0 4273, allowing a biconditional, and of disj2 4249. This proof does not rely, even indirectly, on ssdifin0 4273 nor disj2 4249. (Contributed by BJ, 11-Nov-2021.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
Theorem	bj-sscon 33586	Contraposition law for relative subsets. Relative and generalized version of ssconb 3965, which it can shorten, as well as conss2 39594. (Contributed by BJ, 11-Nov-2021.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴))
Theorem	bj-vjust2 33587	Justification theorem for bj-df-v 33588. See also vjust 3398 and bj-vjust 33363. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Theorem	bj-df-v 33588	Alternate definition of the universal class. Actually, the current definition df-v 3399 should be proved from this one, and vex 3400 should be proved from this proposed definition together with bj-vexwv 33426, which would remove from vex 3400 dependency on ax-13 2333 (see also comment of bj-vexw 33424). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	bj-df-nul 33589	Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = {𝑥 ∣ ⊥}
Theorem	bj-nul 33590*	Two formulations of the axiom of the empty set ax-nul 5025. Proposal: place it right before ax-nul 5025. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliota 33591*	Definition of the empty set using the definite description binder. See also bj-nuliotaALT 33592. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliotaALT 33592*	Alternate proof of bj-nuliota 33591. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6114). 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 33593	Alternate proof of vtoclgf 3464. Proof from vtoclgft 3455. (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-pwcfsdom 33594	Remove hypothesis from pwcfsdom 9740. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9740.) (Contributed by BJ, 14-Sep-2019.)
⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Theorem	bj-grur1 33595	Remove hypothesis from grur1 9977. 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 33596*	The extension of a predicate is included in a set if and only if it is a set. Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5017. 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 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.) TODO: move in place of bm1.3ii 5020.
⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
20.14.5.16  Evaluation
Theorem	bj-evaleq 33597	Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16260 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 33598	The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
⊢ Fun Slot 𝐴
Theorem	bj-evalfn 33599	The evaluation at a class is a function on the universal class. (General form of slotfn 16273). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
⊢ Slot 𝐴 Fn V
Theorem	bj-evalval 33600	Value of the evaluation at a class. (Closed form of strfvnd 16274 and strfvn 16277). (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))