P. 359 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-unrab 35801*	Generalization of unrab 4305. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
Theorem	bj-inrab 35802	Generalization of inrab 4306. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐵 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab2 35803	Shorter proof of inrab 4306. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab3 35804*	Generalization of dfrab3ss 4312, 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 35805*	Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrALT 35806*	Alternate proof of bj-rabtr 35805. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrAUTO 35807*	Proof of bj-rabtr 35805 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
21.17.5.8  Generalized class abstractions
Syntax	bj-cgab 35808	Syntax for generalized class abstractions.
class {𝐴 ∣ 𝑥 ∣ 𝜑}
Definition	df-bj-gab 35809*	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 35810	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 35811	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 35812	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 35813*	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 35814	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 35815	Generalized class abstraction as a direct image. TODO: improve the support lemmas elimag 6063 and fvelima 6957 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}))
21.17.5.9  Restricted nonfreeness In this subsection, we define restricted nonfreeness (or relative nonfreeness).
Syntax	wrnf 35816	Syntax for restricted nonfreeness.
wff Ⅎ𝑥 ∈ 𝐴𝜑
Definition	df-bj-rnf 35817	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))
21.17.5.10  Russell's paradox A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 35818) and then two versions (bj-ru1 35819 and bj-ru 35820). Special attention is put on minimizing axiom depencencies.
Theorem	bj-ru0 35818*	The FOL part of Russell's paradox ru 3776 (see also bj-ru1 35819, bj-ru 35820). Use of elequ1 2113, bj-elequ12 35551 (instead of eleq1 2821, eleq12d 2827 as in ru 3776) permits to remove dependency on ax-10 2137, ax-11 2154, ax-12 2171, ax-ext 2703, df-sb 2068, df-clab 2710, df-cleq 2724, df-clel 2810. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
Theorem	bj-ru1 35819*	A version of Russell's paradox ru 3776 (see also bj-ru 35820). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
Theorem	bj-ru 35820	Remove dependency on ax-13 2371 (and df-v 3476) from Russell's paradox ru 3776 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2814 instead of isset 3487 to avoid use of df-v 3476. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉
21.17.5.11  Curry's paradox in set theory
Theorem	currysetlem 35821*	Lemma for currysetlem 35821, 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 35822*	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 35826. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
Theorem	currysetlem1 35823*	Lemma for currysetALT 35826. (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 35824*	Lemma for currysetALT 35826. (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 35825*	Lemma for currysetALT 35826. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ ¬ 𝑋 ∈ 𝑉
Theorem	currysetALT 35826*	Alternate proof of curryset 35822, 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
21.17.5.12  Some disjointness results A few utility theorems on disjointness of classes.
Theorem	bj-n0i 35827*	Inference associated with n0 4346. Shortens 2ndcdisj 22959 (2888>2878), notzfaus 5361 (264>253). (Contributed by BJ, 22-Apr-2019.)
⊢ 𝐴 ≠ ∅    ⇒   ⊢ ∃𝑥 𝑥 ∈ 𝐴
Theorem	bj-disjsn01 35828	Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9598 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
⊢ ({∅} ∩ {1o}) = ∅
Theorem	bj-0nel1 35829	The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∉ {1o}
Theorem	bj-1nel0 35830	1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
⊢ 1o ∉ {∅}
21.17.5.13  Complements on direct products A few utility theorems on direct products.
Theorem	bj-xpimasn 35831	The image of a singleton, general case. [Change and relabel xpimasn 6184 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
Theorem	bj-xpima1sn 35832	The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6184 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima1snALT 35833	Alternate proof of bj-xpima1sn 35832. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima2sn 35834	The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6184.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	bj-xpnzex 35835	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 7910 (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 35836	Curried (exported) form of xpexg 7736. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V))
Theorem	bj-xpnzexb 35837	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 35838*	Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
21.17.5.14  "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 35839*	The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5285. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Theorem	bj-clexab 35840*	Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
Syntax	bj-csngl 35841	Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
Definition	df-bj-sngl 35842*	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 35843	Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Theorem	bj-elsngl 35844*	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 35845	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 35846	The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-0nelsngl 35847	The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8465). (Contributed by BJ, 6-Oct-2018.)
⊢ ∅ ∉ sngl 𝐴
Theorem	bj-snglinv 35848*	Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
Theorem	bj-snglex 35849	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 35850	Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
Definition	df-bj-tag 35851	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 35852	Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
Theorem	bj-eltag 35853*	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 35854	The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∈ tag 𝐴
Theorem	bj-tagn0 35855	The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ tag 𝐴 ≠ ∅
Theorem	bj-tagss 35856	The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-snglsstag 35857	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ tag 𝐴
Theorem	bj-sngltagi 35858	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵)
Theorem	bj-sngltag 35859	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 35860	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 35861	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 35862*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Theorem	bj-tagex 35863	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 35864	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 35865	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))
21.17.5.15  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 35897 and bj-2uplex 35898, and more importantly, bj-pr21val 35889 and bj-pr22val 35895. 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 35898 has advantages: in view of df-br 5149, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5803 (resp. relsnop 5805) would hold without antecedents (resp. hypotheses) thanks to relsnb 5802). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5728 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 17079 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 4635) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9612) 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 5740, but here we use "tagged versions" of the factors (see df-bj-tag 35851) 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 35851)
Syntax	bj-cproj 35866	Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
Definition	df-bj-proj 35867*	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 35868	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Theorem	bj-projeq2 35869	Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
Theorem	bj-projun 35870	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 35871	Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)
Theorem	bj-projval 35872	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 35873	Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class ⦅𝐴⦆
Definition	df-bj-1upl 35874	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 35888, bj-2uplth 35897, bj-2uplex 35898, and the properties of the projections (see df-bj-pr1 35877 and df-bj-pr2 35891). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
Theorem	bj-1upleq 35875	Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
Syntax	bj-cpr1 35876	Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
Definition	df-bj-pr1 35877	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 35878, bj-pr11val 35881, bj-pr21val 35889, bj-pr1ex 35882. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
⊢ pr1 𝐴 = (∅ Proj 𝐴)
Theorem	bj-pr1eq 35878	Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Theorem	bj-pr1un 35879	The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Theorem	bj-pr1val 35880	Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Theorem	bj-pr11val 35881	Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
⊢ pr1 ⦅𝐴⦆ = 𝐴
Theorem	bj-pr1ex 35882	Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V)
Theorem	bj-1uplth 35883	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 35884	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 35885	A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ ⦅𝐴⦆ ≠ ∅
Syntax	bj-c2uple 35886	Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class ⦅𝐴, 𝐵⦆
Definition	df-bj-2upl 35887	Definition of the Morse couple. See df-bj-1upl 35874. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 35888, bj-2uplth 35897, bj-2uplex 35898, and the properties of the projections (see df-bj-pr1 35877 and df-bj-pr2 35891). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
Theorem	bj-2upleq 35888	Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Theorem	bj-pr21val 35889	Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
Syntax	bj-cpr2 35890	Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
Definition	df-bj-pr2 35891	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 35892, bj-pr22val 35895, bj-pr2ex 35896. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ pr2 𝐴 = (1o Proj 𝐴)
Theorem	bj-pr2eq 35892	Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Theorem	bj-pr2un 35893	The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Theorem	bj-pr2val 35894	Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Theorem	bj-pr22val 35895	Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
Theorem	bj-pr2ex 35896	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 35897	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5476). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 35898	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 35899	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 35900	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 35885 and bj-2upln0 35899 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⦆