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Theorem List for Metamath Proof Explorer - 33501-33600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-rabtrALT 33501* Alternate proof of bj-rabtr 33500. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrAUTO 33502* Proof of bj-rabtr 33500 found automatically by "MM-PA> IMPROVE ALL / DEPTH 3 / 3" followed by "MM-PA> MINIMIZEWITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

20.14.5.9  Restricted non-freeness

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

Syntaxwrnf 33503 Syntax for restricted non-freeness.
wff 𝑥𝐴𝜑

Definitiondf-bj-rnf 33504 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))

A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 33505) and then two versions (bj-ru1 33506 and bj-ru 33507). Special attention is put on minimizing axiom depencencies.

Theorembj-ru0 33505* The FOL part of Russell's paradox ru 3650 (see also bj-ru1 33506, bj-ru 33507). Use of elequ1 2113, bj-elequ12 33257, bj-spvv 33311 (instead of eleq1 2846, eleq12d 2852, spv 2357 as in ru 3650) permits to remove dependency on ax-10 2134, ax-11 2149, ax-12 2162, ax-13 2333, ax-ext 2753, df-sb 2012, df-clab 2763, df-cleq 2769, df-clel 2773. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)

Theorembj-ru1 33506* A version of Russell's paradox ru 3650 (see also bj-ru 33507). Note the more economical use of bj-abeq2 33350 instead of abeq2 2891 to avoid dependency on ax-13 2333. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}

Theorembj-ru 33507 Remove dependency on ax-13 2333 (and df-v 3399) from Russell's paradox ru 3650 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 5018 does require ax-8 2108 and ax-9 2115 since it requires df-clel 2773 and df-cleq 2769--- see bj-df-clel 33459 and bj-df-cleq 33464). Note the more economical use of bj-elissetv 33431 instead of isset 3408 to avoid use of df-v 3399. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉

20.14.5.11  Some disjointness results

A few utility theorems on disjointness of classes.

Theorembj-n0i 33508* Inference associated with n0 4158. Shortens 2ndcdisj 21668 (2888>2878), notzfaus 5074 (264>253). (Contributed by BJ, 22-Apr-2019.)
𝐴 ≠ ∅       𝑥 𝑥𝐴

Theorembj-disjcsn 33509 A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 31405 and does not depend on df-ne 2969. (Contributed by BJ, 4-Apr-2019.)
(𝐴 ∩ {𝐴}) = ∅

Theorembj-disjsn01 33510 Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33509 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
({∅} ∩ {1o}) = ∅

Theorembj-2ex 33511 2o is a set. (Contributed by BJ, 6-Apr-2019.)
2o ∈ V

Theorembj-0nel1 33512 The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
∅ ∉ {1o}

Theorembj-1nel0 33513 1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
1o ∉ {∅}

20.14.5.12  Complements on direct products

A few utility theorems on direct products.

Theorembj-xpimasn 33514 The image of a singleton, general case. [Change and relabel xpimasn 5833 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)

Theorembj-xpima1sn 33515 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5833 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima1snALT 33516 Alternate proof of bj-xpima1sn 33515. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima2sn 33517 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5833] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theorembj-xpnzex 33518 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 7387 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉𝐵 ∈ V))

Theorembj-xpexg2 33519 Curried (exported) form of xpexg 7237. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × 𝐵) ∈ V))

Theorembj-xpnzexb 33520 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 33521* Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})

20.14.5.13  "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 33522* 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 33523* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5006. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

Theorembj-clex 33524* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐵)} ∈ V)

Syntaxbj-csngl 33525 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴

Definitiondf-bj-sngl 33526* 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 33527 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)

Theorembj-elsngl 33528* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})

Theorembj-snglc 33529 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 33530 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ 𝒫 𝐴

Theorembj-0nelsngl 33531 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7843). (Contributed by BJ, 6-Oct-2018.)
∅ ∉ sngl 𝐴

Theorembj-snglinv 33532* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Theorembj-snglex 33533 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 33534 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴

Definitiondf-bj-tag 33535 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 33536 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Theorembj-eltag 33537* 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 33538 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
∅ ∈ tag 𝐴

Theorembj-tagn0 33539 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
tag 𝐴 ≠ ∅

Theorembj-tagss 33540 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
tag 𝐴 ⊆ 𝒫 𝐴

Theorembj-snglsstag 33541 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ tag 𝐴

Theorembj-sngltagi 33542 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵𝐴 ∈ tag 𝐵)

Theorembj-sngltag 33543 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 33544 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 33545 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 33546* Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Theorembj-tagex 33547 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 33548 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 33549 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))

20.14.5.14  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 33581 and bj-2uplex 33582 (but more importantly, bj-pr21val 33573 and bj-pr22val 33579). 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 33582 has advantages: in view of df-br 4887, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5474 (resp. relsnop 5476) would hold without antecedents (resp. hypotheses) thanks to relsnb 5473). Similarly, df-struct 16257 could be simplified with the exception of the empty set removed.

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 4404) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 8810) 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 5413, but here we use "tagged versions" of the factors (see df-bj-tag 33535) 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 33550 Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)

Definitiondf-bj-proj 33551* 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 33552 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))

Theorembj-projeq2 33553 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))

Theorembj-projun 33554 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 33555 Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)

Theorembj-projval 33556 Value of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntaxbj-c1upl 33557 Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class 𝐴

Definitiondf-bj-1upl 33558 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 33572, bj-2uplth 33581, bj-2uplex 33582, and the properties of the projections (see df-bj-pr1 33561 and df-bj-pr2 33575). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
𝐴⦆ = ({∅} × tag 𝐴)

Theorembj-1upleq 33559 Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)

Syntaxbj-cpr1 33560 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴

Definitiondf-bj-pr1 33561 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 33562, bj-pr11val 33565, bj-pr21val 33573, bj-pr1ex 33566. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
pr1 𝐴 = (∅ Proj 𝐴)

Theorembj-pr1eq 33562 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Theorembj-pr1un 33563 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Theorembj-pr1val 33564 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)

Theorembj-pr11val 33565 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
pr1𝐴⦆ = 𝐴

Theorembj-pr1ex 33566 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr1 𝐴 ∈ V)

Theorembj-1uplth 33567 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Theorembj-1uplex 33568 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 33569 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
𝐴⦆ ≠ ∅

Syntaxbj-c2uple 33570 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class 𝐴, 𝐵

Definitiondf-bj-2upl 33571 Definition of the Morse couple. See df-bj-1upl 33558. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 33572, bj-2uplth 33581, bj-2uplex 33582, and the properties of the projections (see df-bj-pr1 33561 and df-bj-pr2 33575). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))

Theorembj-2upleq 33572 Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Theorembj-pr21val 33573 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr1𝐴, 𝐵⦆ = 𝐴

Syntaxbj-cpr2 33574 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴

Definitiondf-bj-pr2 33575 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 33576, bj-pr22val 33579, bj-pr2ex 33580. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
pr2 𝐴 = (1o Proj 𝐴)

Theorembj-pr2eq 33576 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Theorembj-pr2un 33577 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Theorembj-pr2val 33578 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)

Theorembj-pr22val 33579 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr2𝐴, 𝐵⦆ = 𝐵

Theorembj-pr2ex 33580 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)

Theorembj-2uplth 33581 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5176). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembj-2uplex 33582 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 33583 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅

Theorembj-2upln1upl 33584 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 33569 and bj-2upln0 33583 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

20.14.5.15  Set theory: miscellaneous

Miscellaneous theorems of set theory.

Theorembj-disj2r 33585 Relative version of ssdifin0 4273, allowing a biconditional, and of disj2 4249. This proof does not rely, even indirectly, on ssdifin0 4273 nor disj2 4249. (Contributed by BJ, 11-Nov-2021.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Theorembj-sscon 33586 Contraposition law for relative subsets. Relative and generalized version of ssconb 3965, which it can shorten, as well as conss2 39594. (Contributed by BJ, 11-Nov-2021.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Theorembj-vjust2 33587 Justification theorem for bj-df-v 33588. See also vjust 3398 and bj-vjust 33363. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Theorembj-df-v 33588 Alternate definition of the universal class. Actually, the current definition df-v 3399 should be proved from this one, and vex 3400 should be proved from this proposed definition together with bj-vexwv 33426, which would remove from vex 3400 dependency on ax-13 2333 (see also comment of bj-vexw 33424). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}

Theorembj-df-nul 33589 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ⊥}

Theorembj-nul 33590* Two formulations of the axiom of the empty set ax-nul 5025. Proposal: place it right before ax-nul 5025. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliota 33591* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 33592. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliotaALT 33592* Alternate proof of bj-nuliota 33591. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6114). 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 33593 Alternate proof of vtoclgf 3464. Proof from vtoclgft 3455. (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-pwcfsdom 33594 Remove hypothesis from pwcfsdom 9740. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9740.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Theorembj-grur1 33595 Remove hypothesis from grur1 9977. 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 33596* The extension of a predicate is included in a set if and only if it is a set. Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5017. 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 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.)

TODO: move in place of bm1.3ii 5020.

(∃𝑥𝑧(𝜑𝑧𝑥) ↔ ∃𝑦𝑧(𝑧𝑦𝜑))

20.14.5.16  Evaluation

Theorembj-evaleq 33597 Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16260 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 33598 The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
Fun Slot 𝐴

Theorembj-evalfn 33599 The evaluation at a class is a function on the universal class. (General form of slotfn 16273). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
Slot 𝐴 Fn V

Theorembj-evalval 33600 Value of the evaluation at a class. (Closed form of strfvnd 16274 and strfvn 16277). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
(𝐹𝑉 → (Slot 𝐴𝐹) = (𝐹𝐴))

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