P. 370 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-inrab2 36901	Shorter proof of inrab 4269. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab3 36902*	Generalization of dfrab3ss 4276, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵)
Theorem	bj-rabtr 36903*	Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrALT 36904*	Alternate proof of bj-rabtr 36903. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrAUTO 36905*	Proof of bj-rabtr 36903 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 36906	Syntax for generalized class abstractions.
class {𝐴 ∣ 𝑥 ∣ 𝜑}
Definition	df-bj-gab 36907*	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 36908	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 36909	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 36910	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 36911*	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 36912	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 36913	Generalized class abstraction as a direct image. TODO: improve the support lemmas elimag 6019 and fvelima 6892 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 36914	Syntax for restricted nonfreeness.
wff Ⅎ𝑥 ∈ 𝐴𝜑
Definition	df-bj-rnf 36915	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 2128) and then two versions (bj-ru1 36916 and bj-ru 36917). Special attention is put on minimizing axiom depencencies.
Theorem	bj-ru1 36916*	A version of Russell's paradox ru 3742 not mentioning the universal class. (see also bj-ru 36917). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2142, ax-11 2158, ax-12 2178 by using eqabbw 2802 following BTernaryTau's similar revision of ru 3742. (Revised by BJ, 28-Jun-2025.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
Theorem	bj-ru 36917	Remove dependency on ax-13 2370 (and df-v 3440) from Russell's paradox ru 3742 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2809 instead of isset 3452 to avoid use of df-v 3440. (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 36918*	Lemma for currysetlem 36918, 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 36919*	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 36923. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
Theorem	currysetlem1 36920*	Lemma for currysetALT 36923. (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 36921*	Lemma for currysetALT 36923. (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 36922*	Lemma for currysetALT 36923. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ ¬ 𝑋 ∈ 𝑉
Theorem	currysetALT 36923*	Alternate proof of curryset 36919, 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 36924*	Inference associated with n0 4306. Shortens 2ndcdisj 23359 (2888>2878), notzfaus 5305 (264>253). (Contributed by BJ, 22-Apr-2019.)
⊢ 𝐴 ≠ ∅    ⇒   ⊢ ∃𝑥 𝑥 ∈ 𝐴
Theorem	bj-disjsn01 36925	Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9518 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
⊢ ({∅} ∩ {1o}) = ∅
Theorem	bj-0nel1 36926	The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∉ {1o}
Theorem	bj-1nel0 36927	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 36928	The image of a singleton, general case. [Change and relabel xpimasn 6138 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
Theorem	bj-xpima1sn 36929	The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6138 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima1snALT 36930	Alternate proof of bj-xpima1sn 36929. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima2sn 36931	The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6138.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	bj-xpnzex 36932	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 7860 (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 36933	Curried (exported) form of xpexg 7690. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V))
Theorem	bj-xpnzexb 36934	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 36935*	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 36936*	The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5221. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Theorem	bj-clexab 36937*	Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
Syntax	bj-csngl 36938	Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
Definition	df-bj-sngl 36939*	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 36940	Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Theorem	bj-elsngl 36941*	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 36942	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 36943	The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-0nelsngl 36944	The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8395). (Contributed by BJ, 6-Oct-2018.)
⊢ ∅ ∉ sngl 𝐴
Theorem	bj-snglinv 36945*	Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
Theorem	bj-snglex 36946	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 36947	Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
Definition	df-bj-tag 36948	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 36949	Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
Theorem	bj-eltag 36950*	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 36951	The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∈ tag 𝐴
Theorem	bj-tagn0 36952	The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ tag 𝐴 ≠ ∅
Theorem	bj-tagss 36953	The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-snglsstag 36954	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ tag 𝐴
Theorem	bj-sngltagi 36955	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵)
Theorem	bj-sngltag 36956	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 36957	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 36958	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 36959*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Theorem	bj-tagex 36960	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 36961	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 36962	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 36994 and bj-2uplex 36995, and more importantly, bj-pr21val 36986 and bj-pr22val 36992. 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 36995 has advantages: in view of df-br 5096, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5750 (resp. relsnop 5752) would hold without antecedents (resp. hypotheses) thanks to relsnb 5749). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5675 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 17076 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 4586) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9533) 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 5687, but here we use "tagged versions" of the factors (see df-bj-tag 36948) 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 36948)
Syntax	bj-cproj 36963	Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
Definition	df-bj-proj 36964*	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 36965	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Theorem	bj-projeq2 36966	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
Theorem	bj-projun 36967	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 36968	Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)
Theorem	bj-projval 36969	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 36970	Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class ⦅𝐴⦆
Definition	df-bj-1upl 36971	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 36985, bj-2uplth 36994, bj-2uplex 36995, and the properties of the projections (see df-bj-pr1 36974 and df-bj-pr2 36988). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
Theorem	bj-1upleq 36972	Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
Syntax	bj-cpr1 36973	Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
Definition	df-bj-pr1 36974	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 36975, bj-pr11val 36978, bj-pr21val 36986, bj-pr1ex 36979. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ pr1 𝐴 = (∅ Proj 𝐴)
Theorem	bj-pr1eq 36975	Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Theorem	bj-pr1un 36976	The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Theorem	bj-pr1val 36977	Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Theorem	bj-pr11val 36978	Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ⦅𝐴⦆ = 𝐴
Theorem	bj-pr1ex 36979	Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V)
Theorem	bj-1uplth 36980	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 36981	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 36982	A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ ⦅𝐴⦆ ≠ ∅
Syntax	bj-c2uple 36983	Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class ⦅𝐴, 𝐵⦆
Definition	df-bj-2upl 36984	Definition of the Morse couple. See df-bj-1upl 36971. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 36985, bj-2uplth 36994, bj-2uplex 36995, and the properties of the projections (see df-bj-pr1 36974 and df-bj-pr2 36988). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
Theorem	bj-2upleq 36985	Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Theorem	bj-pr21val 36986	Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
Syntax	bj-cpr2 36987	Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
Definition	df-bj-pr2 36988	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 36989, bj-pr22val 36992, bj-pr2ex 36993. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ pr2 𝐴 = (1o Proj 𝐴)
Theorem	bj-pr2eq 36989	Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Theorem	bj-pr2un 36990	The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Theorem	bj-pr2val 36991	Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Theorem	bj-pr22val 36992	Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
Theorem	bj-pr2ex 36993	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 36994	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5423). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 36995	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 36996	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 36997	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 36982 and bj-2upln0 36996 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 36998	Relative version of cleqf 2920. (Contributed by BJ, 27-Dec-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑉    ⇒   ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-rcleq 36999*	Relative version of dfcleq 2722. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-reabeq 37000*	Relative form of eqabb 2867. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))