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Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-0nel1 34301 The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
∅ ∉ {1o}

Theorembj-1nel0 34302 1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
1o ∉ {∅}

20.15.5.13  Complements on direct products

A few utility theorems on direct products.

Theorembj-xpimasn 34303 The image of a singleton, general case. [Change and relabel xpimasn 6029 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)

Theorembj-xpima1sn 34304 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6029 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima1snALT 34305 Alternate proof of bj-xpima1sn 34304. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima2sn 34306 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6029.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theorembj-xpnzex 34307 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 7615 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉𝐵 ∈ V))

Theorembj-xpexg2 34308 Curried (exported) form of xpexg 7463. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × 𝐵) ∈ V))

Theorembj-xpnzexb 34309 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 34310* Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})

20.15.5.14  "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.

Theorembj-snsetex 34311* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5176. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

Theorembj-clex 34312* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐵)} ∈ V)

Syntaxbj-csngl 34313 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴

Definitiondf-bj-sngl 34314* 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 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}

Theorembj-sngleq 34315 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)

Theorembj-elsngl 34316* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})

Theorembj-snglc 34317 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 34318 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ 𝒫 𝐴

Theorembj-0nelsngl 34319 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8092). (Contributed by BJ, 6-Oct-2018.)
∅ ∉ sngl 𝐴

Theorembj-snglinv 34320* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Theorembj-snglex 34321 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 34322 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴

Definitiondf-bj-tag 34323 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 34324 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Theorembj-eltag 34325* 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 34326 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
∅ ∈ tag 𝐴

Theorembj-tagn0 34327 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
tag 𝐴 ≠ ∅

Theorembj-tagss 34328 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
tag 𝐴 ⊆ 𝒫 𝐴

Theorembj-snglsstag 34329 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ tag 𝐴

Theorembj-sngltagi 34330 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵𝐴 ∈ tag 𝐵)

Theorembj-sngltag 34331 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 34332 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 34333 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 34334* Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Theorembj-tagex 34335 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 34336 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 34337 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))

20.15.5.15  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 34369 and bj-2uplex 34370, and more importantly, bj-pr21val 34361 and bj-pr22val 34367. 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 34370 has advantages: in view of df-br 5053, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5663 (resp. relsnop 5665) would hold without antecedents (resp. hypotheses) thanks to relsnb 5662). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5591 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 16481 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 4556) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9072) 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 5603, but here we use "tagged versions" of the factors (see df-bj-tag 34323) 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 34323)

Syntaxbj-cproj 34338 Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)

Definitiondf-bj-proj 34339* 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 34340 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))

Theorembj-projeq2 34341 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))

Theorembj-projun 34342 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 34343 Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)

Theorembj-projval 34344 Value of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntaxbj-c1upl 34345 Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class 𝐴

Definitiondf-bj-1upl 34346 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 34360, bj-2uplth 34369, bj-2uplex 34370, and the properties of the projections (see df-bj-pr1 34349 and df-bj-pr2 34363). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
𝐴⦆ = ({∅} × tag 𝐴)

Theorembj-1upleq 34347 Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)

Syntaxbj-cpr1 34348 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴

Definitiondf-bj-pr1 34349 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 34350, bj-pr11val 34353, bj-pr21val 34361, bj-pr1ex 34354. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
pr1 𝐴 = (∅ Proj 𝐴)

Theorembj-pr1eq 34350 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Theorembj-pr1un 34351 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Theorembj-pr1val 34352 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)

Theorembj-pr11val 34353 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
pr1𝐴⦆ = 𝐴

Theorembj-pr1ex 34354 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr1 𝐴 ∈ V)

Theorembj-1uplth 34355 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Theorembj-1uplex 34356 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 34357 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
𝐴⦆ ≠ ∅

Syntaxbj-c2uple 34358 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class 𝐴, 𝐵

Definitiondf-bj-2upl 34359 Definition of the Morse couple. See df-bj-1upl 34346. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 34360, bj-2uplth 34369, bj-2uplex 34370, and the properties of the projections (see df-bj-pr1 34349 and df-bj-pr2 34363). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))

Theorembj-2upleq 34360 Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Theorembj-pr21val 34361 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr1𝐴, 𝐵⦆ = 𝐴

Syntaxbj-cpr2 34362 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴

Definitiondf-bj-pr2 34363 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 34364, bj-pr22val 34367, bj-pr2ex 34368. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
pr2 𝐴 = (1o Proj 𝐴)

Theorembj-pr2eq 34364 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Theorembj-pr2un 34365 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Theorembj-pr2val 34366 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)

Theorembj-pr22val 34367 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr2𝐴, 𝐵⦆ = 𝐵

Theorembj-pr2ex 34368 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)

Theorembj-2uplth 34369 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5355). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembj-2uplex 34370 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))

Theorembj-2upln0 34371 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅

Theorembj-2upln1upl 34372 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 34357 and bj-2upln0 34371 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

20.15.5.16  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).

Theorembj-rcleqf 34373 Relative version of cleqf 3010. (Contributed by BJ, 27-Dec-2023.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑉       ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))

Theorembj-rcleq 34374* Relative version of dfcleq 2818. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))

Theorembj-reabeq 34375* Relative form of abeq2 2948. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))

Theorembj-disj2r 34376 Relative version of ssdifin0 4413, allowing a biconditional, and of disj2 4389. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4413 nor disj2 4389. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Theorembj-sscon 34377 Contraposition law for relative subclasses. Relative and generalized version of ssconb 4099, which it can shorten, as well as conss2 41004. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4099 nor conss2 41004. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

20.15.5.17  Set theory: miscellaneous

Miscellaneous theorems of set theory.

Theorembj-pw0ALT 34378 Alternate proof of pw0 4729. The proofs have a similar structure: pw0 4729 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 34378 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4729 and biconditional for bj-pw0ALT 34378) to translate the property ss0b 4333 into the wanted result. To translate a biconditional into a class equality, pw0 4729 uses abbii 2889 (which yields an equality of class abstractions), while bj-pw0ALT 34378 uses eqriv 2821 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2889, through its closed form abbi1 2887, is proved from eqrdv 2822, which is the deduction form of eqriv 2821. In the other direction, velpw 4526 and velsn 4565 are proved from the definitions of powerclass and singleton using elabg 3652, which is a version of abbii 2889 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 ∅ = {∅}

Theorembj-sselpwuni 34379 Quantitative version of ssexg 5213: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.)
((𝐴𝐵𝐵𝑉) → 𝐴 ∈ 𝒫 𝑉)

Theorembj-unirel 34380 Quantitative version of uniexr 7475: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)
( 𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝑉)

Theorembj-elpwg 34381 If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4524 and elpw2g 5233 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theorembj-vjust 34382 Justification theorem for bj-df-v 34383. See also vjust 3481. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Theorembj-df-v 34383 Alternate definition of the universal class. Actually, the current definition df-v 3482 should be proved from this one, and vex 3483 should be proved from this proposed definition together with vexw 2808, which would remove from vex 3483 dependency on ax-13 2392 (see also comment of vexw 2808). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}

Theorembj-df-nul 34384 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ⊥}

Theorembj-nul 34385* Two formulations of the axiom of the empty set ax-nul 5196. Proposal: place it right before ax-nul 5196. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliota 34386* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 34387. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliotaALT 34387* Alternate proof of bj-nuliota 34386. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6321). 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.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-vtoclgfALT 34388 Alternate proof of vtoclgf 3551. Proof from vtoclgft 3539. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-elsn12g 34389 Join of elsng 4563 and elsn2g 4587. (Contributed by BJ, 18-Nov-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Theorembj-elsnb 34390 Biconditional version of elsng 4563. (Contributed by BJ, 18-Nov-2023.)
(𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Theorembj-pwcfsdom 34391 Remove hypothesis from pwcfsdom 9997. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9997.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))

Theorembj-grur1 34392 Remove hypothesis from grur1 10234. 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)))

Theorembj-bm1.3ii 34393* The extension of a predicate (𝜑(𝑧)) is included in a set (𝑥) if and only if it is a set (𝑦). Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5189. 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 𝑥, 𝑦 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.)

TODO: move in place of bm1.3ii 5192. Relabel ("sepbi"?).

(∃𝑥𝑧(𝜑𝑧𝑥) ↔ ∃𝑦𝑧(𝑧𝑦𝜑))

Theorembj-0nelopab 34394 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by BJ, 22-Jul-2023.)

TODO: move to the main section when one can reorder sections so that we can use relopab 5683 (this is a very limited reordering).

¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theorembj-brrelex12ALT 34395 Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5591. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-epelg 34396 The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5456 and closed form of epeli 5455. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5595 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Theorembj-epelb 34397 Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5596 available. Check if it is shorter to prove bj-epelg 34396 first or bj-epelb 34397 first. (Contributed by BJ, 14-Jul-2023.)
(𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))

Theorembj-nsnid 34398 A set does not contain the singleton formed on it. More precisely, one can prove that a class contains the singleton formed on it if and only if it is proper and contains the empty set (since it is "the singleton formed on" any proper class, see snprc 4637): ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
(𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)

20.15.5.18  Evaluation

This section treats the existing predicate Slot (df-slot 16483) as "evaluation at a class" and for the moment does not introduce new syntax for it.

Theorembj-evaleq 34399 Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16484 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)

Theorembj-evalfun 34400 The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
Fun Slot 𝐴

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