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Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-rabeqbid 32901 Version of rabeqbidv 3193 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theorembj-rabeqbida 32902 Version of rabeqbidva 3194 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theorembj-seex 32903* Version of seex 5075 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
𝑥𝐵       ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
 
Theorembj-nfcf 32904* Version of df-nfc 2752 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
𝑦𝐴       (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
 
Theorembj-axsep2 32905* Remove dependency on ax-12 2046 and ax-13 2245 from axsep2 4780 while shortening its proof. Remark: the comment in zfauscl 4781 is misleading: the essential use of ax-ext 2601 is the one via eleq2 2689 and not the one via vtocl 3257, since the latter can be proved without ax-ext 2601 (see bj-vtocl 32893). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
20.14.5.7  Class abstractions

A few additional theorems on class abstractions and restricted class abstractions.

 
Theorembj-unrab 32906* Generalization of unrab 3896. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
 
Theorembj-inrab 32907 Generalization of inrab 3897. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
 
Theorembj-inrab2 32908 Shorter proof of inrab 3897. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theorembj-inrab3 32909* Generalization of dfrab3ss 3903, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
(𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)
 
Theorembj-rabtr 32910* Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
{𝑥𝐴 ∣ ⊤} = 𝐴
 
Theorembj-rabtrALT 32911* Alternate proof of bj-rabtr 32910. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴
 
Theorembj-rabtrALTALT 32912* Alternate proof of bj-rabtr 32910. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴
 
Theorembj-rabtrAUTO 32913* Proof of bj-rabtr 32910 found automatically by "improve all /depth 3 /3" followed by "minimize *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴
 
20.14.5.8  Restricted non-freeness

In this subsection, we define restricted non-freeness (or relative non-freeness).

 
Syntaxwrnf 32914 Syntax for restricted non-freeness.
wff 𝑥𝐴𝜑
 
Definitiondf-bj-rnf 32915 Definition of restricted non-freeness. Informally, the proposition 𝑥𝐴𝜑 means that 𝜑(𝑥) does not vary on 𝐴. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝐴𝜑 ↔ (∃𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜑))
 
20.14.5.9  Russell's paradox

A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 32916) and then two versions (bj-ru1 32917 and bj-ru 32918). Special attention is put on minimizing axiom depencencies.

 
Theorembj-ru0 32916* The FOL part of Russell's paradox ru 3432 (see also bj-ru1 32917, bj-ru 32918). Use of elequ1 1996, bj-elequ12 32652, bj-spvv 32707 (instead of eleq1 2688, eleq12d 2694, spv 2259 as in ru 3432) permits to remove dependency on ax-10 2018, ax-11 2033, ax-12 2046, ax-13 2245, ax-ext 2601, df-sb 1880, df-clab 2608, df-cleq 2614, df-clel 2617. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
 
Theorembj-ru1 32917* A version of Russell's paradox ru 3432 (see also bj-ru 32918). Note the more economical use of bj-abeq2 32757 instead of abeq2 2731 to avoid dependency on ax-13 2245. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
 
Theorembj-ru 32918 Remove dependency on ax-13 2245 (and df-v 3200) from Russell's paradox ru 3432 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4780 does require ax-8 1991 and ax-9 1998 since it requires df-clel 2617 and df-cleq 2614--- see bj-df-clel 32872 and bj-df-cleq 32877). Note the more economical use of bj-elissetv 32845 instead of isset 3205 to avoid use of df-v 3200. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉
 
20.14.5.10  Some disjointness results

A few utility theorems on disjointness of classes.

 
Theorembj-n0i 32919* Inference associated with n0 3929. Shortens 2ndcdisj 21253 (2888>2878), notzfaus 4838 (264>253). (Contributed by BJ, 22-Apr-2019.)
𝐴 ≠ ∅       𝑥 𝑥𝐴
 
Theorembj-disjcsn 32920 A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 30790 and does not depend on df-ne 2794. (Contributed by BJ, 4-Apr-2019.)
(𝐴 ∩ {𝐴}) = ∅
 
Theorembj-disjsn01 32921 Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 32920 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
({∅} ∩ {1𝑜}) = ∅
 
Theorembj-1ex 32922 1𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
1𝑜 ∈ V
 
Theorembj-2ex 32923 2𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
2𝑜 ∈ V
 
Theorembj-0nel1 32924 The empty set does not belong to {1𝑜}. (Contributed by BJ, 6-Apr-2019.)
∅ ∉ {1𝑜}
 
Theorembj-1nel0 32925 1𝑜 does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
1𝑜 ∉ {∅}
 
20.14.5.11  Complements on direct products

A few utility theorems on direct products.

 
Theorembj-xpimasn 32926 The image of a singleton, general case. [Change and relabel xpimasn 5577 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
 
Theorembj-xpima1sn 32927 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5577 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
 
Theorembj-xpima1snALT 32928 Alternate proof of bj-xpima1sn 32927. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
 
Theorembj-xpima2sn 32929 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5577] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
 
Theorembj-xpnzex 32930 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 7105 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉𝐵 ∈ V))
 
Theorembj-xpexg2 32931 Curried (exported) form of xpexg 6957. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × 𝐵) ∈ V))
 
Theorembj-xpnzexb 32932 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))
 
Theorembj-cleq 32933* Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
 
20.14.5.12  "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.

 
Theorembj-sels 32934* If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
(𝐴𝑉 → ∃𝑥 𝐴𝑥)
 
Theorembj-snsetex 32935* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 4769. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
 
Theorembj-clex 32936* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐵)} ∈ V)
 
Syntaxbj-csngl 32937 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
 
Definitiondf-bj-sngl 32938* Definition of "singletonization". The class sngl 𝐴 is isomorphic to 𝐴 and since it contains only singletons, it can be adjoined disjoint elements, which can be useful in various constructions. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}
 
Theorembj-sngleq 32939 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
 
Theorembj-elsngl 32940* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
 
Theorembj-snglc 32941 Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
 
Theorembj-snglss 32942 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ 𝒫 𝐴
 
Theorembj-0nelsngl 32943 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7557). (Contributed by BJ, 6-Oct-2018.)
∅ ∉ sngl 𝐴
 
Theorembj-snglinv 32944* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
 
Theorembj-snglex 32945 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)
 
Syntaxbj-ctag 32946 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
 
Definitiondf-bj-tag 32947 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 ∪ {∅})
 
Theorembj-tageq 32948 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
 
Theorembj-eltag 32949* Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
 
Theorembj-0eltag 32950 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
∅ ∈ tag 𝐴
 
Theorembj-tagn0 32951 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
tag 𝐴 ≠ ∅
 
Theorembj-tagss 32952 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
tag 𝐴 ⊆ 𝒫 𝐴
 
Theorembj-snglsstag 32953 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ tag 𝐴
 
Theorembj-sngltagi 32954 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵𝐴 ∈ tag 𝐵)
 
Theorembj-sngltag 32955 The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
 
Theorembj-tagci 32956 Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝐵 → {𝐴} ∈ tag 𝐵)
 
Theorembj-tagcg 32957 Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ∈ tag 𝐵))
 
Theorembj-taginv 32958* Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
 
Theorembj-tagex 32959 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)
 
Theorembj-xtageq 32960 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))
 
Theorembj-xtagex 32961 The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × tag 𝐵) ∈ V))
 
20.14.5.13  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 32993 and bj-2uplex 32994 (but more importantly, bj-pr21val 32985 and bj-pr22val 32991). In particular, one can define well-behaved tuples of classes. Note, however, that classes in ZF(C) are only virtual, and in particular they cannot be quantified over.

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 4182) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 8512) 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 5165, but here we use "tagged versions" of the factors (see df-bj-tag 32947) 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)

 
Syntaxbj-cproj 32962 Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)
 
Definitiondf-bj-proj 32963* 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})}
 
Theorembj-projeq 32964 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
 
Theorembj-projeq2 32965 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))
 
Theorembj-projun 32966 The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
(𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
 
Theorembj-projex 32967 Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)
 
Theorembj-projval 32968 Value of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))
 
Syntaxbj-c1upl 32969 Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class 𝐴
 
Definitiondf-bj-1upl 32970 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 32984, bj-2uplth 32993, bj-2uplex 32994, and the properties of the projections (see df-bj-pr1 32973 and df-bj-pr2 32987). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
𝐴⦆ = ({∅} × tag 𝐴)
 
Theorembj-1upleq 32971 Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
 
Syntaxbj-cpr1 32972 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
 
Definitiondf-bj-pr1 32973 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 32974, bj-pr11val 32977, bj-pr21val 32985, bj-pr1ex 32978. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
pr1 𝐴 = (∅ Proj 𝐴)
 
Theorembj-pr1eq 32974 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
 
Theorembj-pr1un 32975 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
 
Theorembj-pr1val 32976 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
 
Theorembj-pr11val 32977 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
pr1𝐴⦆ = 𝐴
 
Theorembj-pr1ex 32978 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr1 𝐴 ∈ V)
 
Theorembj-1uplth 32979 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
 
Theorembj-1uplex 32980 A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
 
Theorembj-1upln0 32981 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
𝐴⦆ ≠ ∅
 
Syntaxbj-c2uple 32982 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class 𝐴, 𝐵
 
Definitiondf-bj-2upl 32983 Definition of the Morse couple. See df-bj-1upl 32970. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 32984, bj-2uplth 32993, bj-2uplex 32994, and the properties of the projections (see df-bj-pr1 32973 and df-bj-pr2 32987). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
 
Theorembj-2upleq 32984 Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
 
Theorembj-pr21val 32985 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr1𝐴, 𝐵⦆ = 𝐴
 
Syntaxbj-cpr2 32986 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
 
Definitiondf-bj-pr2 32987 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 32988, bj-pr22val 32991, bj-pr2ex 32992. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
pr2 𝐴 = (1𝑜 Proj 𝐴)
 
Theorembj-pr2eq 32988 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
 
Theorembj-pr2un 32989 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
 
Theorembj-pr2val 32990 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1𝑜, 𝐵, ∅)
 
Theorembj-pr22val 32991 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr2𝐴, 𝐵⦆ = 𝐵
 
Theorembj-pr2ex 32992 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)
 
Theorembj-2uplth 32993 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4943). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theorembj-2uplex 32994 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))
 
Theorembj-2upln0 32995 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅
 
Theorembj-2upln1upl 32996 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 32981 and bj-2upln0 32995 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
 
20.14.5.14  Set theory: miscellaneous

Miscellaneous theorems of set theory.

 
Theorembj-disj2r 32997 Relative version of ssdifin0 4048, allowing a biconditional, and of disj2 4022. This proof does not rely, even indirectly, on ssdifin0 4048 nor disj2 4022. (Contributed by BJ, 11-Nov-2021.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
 
Theorembj-sscon 32998 Contraposition law for relative subsets. Relative and generalized version of ssconb 3741, which it can shorten, as well as conss2 38473. (Contributed by BJ, 11-Nov-2021.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
 
Theorembj-vjust2 32999 Justification theorem for bj-df-v 33000. See also vjust 3199 and bj-vjust 32770. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
 
Theorembj-df-v 33000 Alternate definition of the universal class. Actually, the current definition df-v 3200 should be proved from this one, and vex 3201 should be proved from this proposed definition together with bj-vexwv 32841, which would remove from vex 3201 dependency on ax-13 2245 (see also comment of bj-vexw 32839). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}
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