P. 375 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37401-37500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-csbsn 37401	Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-sbel1 37402*	Version of sbcel1g 4373 when substituting a set. (Note: one could have a corresponding version of sbcel12 4368 when substituting a set, but the point here is that the antecedent of sbcel1g 4373 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)
Theorem	bj-abv 37403	The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-abvALT 37404	Alternate version of bj-abv 37403; shorter but uses ax-8 2147. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-ab0 37405	The class of sets verifying a falsity is the empty set (closed form of abf 4363). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
Theorem	bj-abf 37406	Shorter proof of abf 4363 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	bj-csbprc 37407	More direct proof of csbprc 4366 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
21.19.5.6  Removing some axiom requirements and disjoint variable conditions
Theorem	bj-exlimvmpi 37408*	A Fol lemma (exlimiv 1953 followed by mpi 21). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpi 37409	Lemma for bj-vtoclg1f1 37414 (an instance of this lemma is a version of bj-vtoclg1f1 37414 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbi 37410	Lemma for theorems of the vtoclg 3525 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbir 37411	Lemma for theorems of the vtoclg 3525 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (∃𝑥𝜒 → 𝜑)
Theorem	bj-vtoclf 37412*	Remove dependency on ax-ext 2737, df-clab 2744 and df-cleq 2757 (and df-sb 2094 and df-v 3459) from vtoclf 3533. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtocl 37413*	Remove dependency on ax-ext 2737, df-clab 2744 and df-cleq 2757 (and df-sb 2094 and df-v 3459) from vtocl 3528. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtoclg1f1 37414*	The FOL content of vtoclg1f 3538 (hence not using ax-ext 2737, df-cleq 2757, df-nfc 2914, df-v 3459). Note the weakened "major" hypothesis and the disjoint variable condition between 𝑥 and 𝐴 (needed since the nonfreeness quantifier for classes is not available without ax-ext 2737; as a byproduct, this dispenses with ax-11 2194 and ax-13 2406). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)
Theorem	bj-vtoclg1f 37415*	Reprove vtoclg1f 3538 from bj-vtoclg1f1 37414. This removes dependency on ax-ext 2737, df-cleq 2757 and df-v 3459. Use bj-vtoclg1fv 37416 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-vtoclg1fv 37416*	Version of bj-vtoclg1f 37415 with a disjoint variable condition on 𝑥, 𝑉. This removes dependency on df-sb 2094 and df-clab 2744. Prefer its use over bj-vtoclg1f 37415 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-vtoclg 37417*	A version of vtoclg 3525 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2744, see bj-vtoclg1f 37415), which requires fewer axioms (i.e., removes dependency on ax-6 1990, ax-7 2031, ax-9 2155, ax-12 2215, ax-ext 2737, df-clab 2744, df-cleq 2757, df-v 3459). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-rabeqbid 37418	Version of rabeqbidv 3435 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	bj-seex 37419*	Version of seex 5611 with a disjoint variable condition replaced by a nonfreeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)
Theorem	bj-nfcf 37420*	Version of df-nfc 2914 with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 2-May-2019.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	bj-zfauscl 37421*	General version of zfauscl 5253. Remark: the comment in zfauscl 5253 is misleading: the essential use of ax-ext 2737 is the one via eleq2 2854 and not the one via vtocl 3528, since the latter can be proved without ax-ext 2737 (see bj-vtoclg 37417). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
21.19.5.7  Class abstractions A few additional theorems on class abstractions and restricted class abstractions.
Theorem	bj-elabd2ALT 37422*	Alternate proof of elabd2 3632 bypassing elab6g 3631 (and using sbiedvw 2132 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
Theorem	bj-unrab 37423*	Generalization of unrab 4270. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
Theorem	bj-inrab 37424	Generalization of inrab 4271. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐵 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab2 37425	Shorter proof of inrab 4271. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab3 37426*	Generalization of dfrab3ss 4278. Shortens dfrab3ss 4278. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵)
Theorem	bj-rabtr 37427*	Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrALT 37428*	Alternate proof of bj-rabtr 37427. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrAUTO 37429*	Proof of bj-rabtr 37427 found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE_WITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
21.19.5.8  Generalized class abstractions
Syntax	bj-cgab 37430	Syntax for generalized class abstractions.
class {𝐴 ∣ 𝑥 ∣ 𝜑}
Definition	df-bj-gab 37431*	Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴 ∣ 𝑥 ∣ 𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.)
⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
Theorem	bj-gabss 37432	Inclusion of generalized class abstractions. (Contributed by BJ, 4-Oct-2024.)
⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓})
Theorem	bj-gabssd 37433	Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
Theorem	bj-gabeqd 37434	Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒})
Theorem	bj-gabeqis 37435*	Equality of generalized class abstractions, with implicit substitution. (Contributed by BJ, 4-Oct-2024.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝐵 ∣ 𝑦 ∣ 𝜓}
Theorem	bj-elgab 37436	Elements of a generalized class abstraction. (Contributed by BJ, 4-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒))
Theorem	bj-gabima 37437	Generalized class abstraction as a direct image. TODO: improve the support lemmas elimag 6057 and fvelima 6936 to nonfreeness hypothesis (and for the latter, biconditional). (Contributed by BJ, 4-Oct-2024.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))
21.19.5.9  Restricted nonfreeness In this subsection, we define restricted nonfreeness (or relative nonfreeness).
Syntax	wrnf 37438	Syntax for restricted nonfreeness.
wff Ⅎ𝑥 ∈ 𝐴𝜑
Definition	df-bj-rnf 37439	Definition of restricted nonfreeness. 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))
21.19.5.10  Russell's paradox A few results around Russell's paradox. For clarity, we prove separately a FOL statement (now in the main part as ru0 2164) and then two versions (bj-ru1 37440 and bj-ru 37441). Special attention is put on minimizing axiom depencencies.
Theorem	bj-ru1 37440*	A version of Russell's paradox ru 3746 not mentioning the universal class. (see also bj-ru 37441). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2178, ax-11 2194, ax-12 2215 by using eqabbw 2838 following BTernaryTau's similar revision of ru 3746. (Revised by BJ, 28-Jun-2025.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
Theorem	bj-ru 37441	Remove dependency on ax-13 2406 (and df-v 3459) from Russell's paradox ru 3746 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2846 instead of isset 3471 to avoid use of df-v 3459. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉
21.19.5.11  Curry's paradox in set theory
Theorem	currysetlem 37442*	Lemma for currysetlem 37442, where it is used with (𝑥 ∈ 𝑥 → 𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))
Theorem	curryset 37443*	Curry's paradox in set theory. This can be seen as a generalization of Russell's paradox, which corresponds to the case where 𝜑 is ⊥. See alternate exposal of basically the same proof currysetALT 37447. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
Theorem	currysetlem1 37444*	Lemma for currysetALT 37447. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
Theorem	currysetlem2 37445*	Lemma for currysetALT 37447. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))
Theorem	currysetlem3 37446*	Lemma for currysetALT 37447. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ ¬ 𝑋 ∈ 𝑉
Theorem	currysetALT 37447*	Alternate proof of curryset 37443, or more precisely alternate exposal of the same proof. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
21.19.5.12  Some disjointness results A few utility theorems on disjointness of classes.
Theorem	bj-n0i 37448*	Inference associated with n0 4308. Shortens 2ndcdisj 23574 (2888>2878), notzfaus 5325 (264>253). (Contributed by BJ, 22-Apr-2019.)
⊢ 𝐴 ≠ ∅    ⇒   ⊢ ∃𝑥 𝑥 ∈ 𝐴
Theorem	bj-disjsn01 37449	Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9560 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
⊢ ({∅} ∩ {1o}) = ∅
Theorem	bj-0nel1 37450	The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∉ {1o}
Theorem	bj-1nel0 37451	1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
⊢ 1o ∉ {∅}
21.19.5.13  Complements on direct products A few utility theorems on direct products.
Theorem	bj-xpimasn 37452	The image of a singleton, general case. [Change and relabel xpimasn 6175 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
Theorem	bj-xpima1sn 37453	The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6175 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima1snALT 37454	Alternate proof of bj-xpima1sn 37453. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima2sn 37455	The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6175.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	bj-xpnzex 37456	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 7905 (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 37457	Curried (exported) form of xpexg 7737. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V))
Theorem	bj-xpnzexb 37458	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 37459*	Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
21.19.5.14  "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-snsetex 37460*	The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5232. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Theorem	bj-clexab 37461*	Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
Syntax	bj-csngl 37462	Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
Definition	df-bj-sngl 37463*	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 37464	Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Theorem	bj-elsngl 37465*	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 37466	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 37467	The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-0nelsngl 37468	The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8441). (Contributed by BJ, 6-Oct-2018.)
⊢ ∅ ∉ sngl 𝐴
Theorem	bj-snglinv 37469*	Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
Theorem	bj-snglex 37470	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 37471	Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
Definition	df-bj-tag 37472	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 37473	Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
Theorem	bj-eltag 37474*	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 37475	The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∈ tag 𝐴
Theorem	bj-tagn0 37476	The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ tag 𝐴 ≠ ∅
Theorem	bj-tagss 37477	The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-snglsstag 37478	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ tag 𝐴
Theorem	bj-sngltagi 37479	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵)
Theorem	bj-sngltag 37480	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 37481	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 37482	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 37483*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Theorem	bj-tagex 37484	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 37485	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 37486	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 37518 and bj-2uplex 37519, and more importantly, bj-pr21val 37510 and bj-pr22val 37516. 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 37519 has advantages: in view of df-br 5106, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5781 (resp. relsnop 5783) would hold without antecedents (resp. hypotheses) thanks to relsnb 5780). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5704 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 17197 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 4592) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9575) 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 5716, but here we use "tagged versions" of the factors (see df-bj-tag 37472) 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 37472)
Syntax	bj-cproj 37487	Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
Definition	df-bj-proj 37488*	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 37489	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Theorem	bj-projeq2 37490	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
Theorem	bj-projun 37491	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 37492	Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)
Theorem	bj-projval 37493	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 37494	Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class ⦅𝐴⦆
Definition	df-bj-1upl 37495	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 37509, bj-2uplth 37518, bj-2uplex 37519, and the properties of the projections (see df-bj-pr1 37498 and df-bj-pr2 37512). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
Theorem	bj-1upleq 37496	Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
Syntax	bj-cpr1 37497	Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
Definition	df-bj-pr1 37498	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 37499, bj-pr11val 37502, bj-pr21val 37510, bj-pr1ex 37503. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ pr1 𝐴 = (∅ Proj 𝐴)
Theorem	bj-pr1eq 37499	Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Theorem	bj-pr1un 37500	The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)