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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-exlimmpbir 34601 Lemma for theorems of the vtoclg 3167 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ph   &    |-  ( ch  ->  (
 ph 
 <->  ps ) )   &    |-  ps   =>    |-  ( E. x ch  ->  ph )
 
Theorembj-vtoclf 34602* Remove dependency on ax-ext 2435, df-clab 2443 and df-cleq 2449 (and df-sb 1741 and df-v 3111) from vtoclf 3160. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  A  e.  V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theorembj-vtocl 34603* Remove dependency on ax-ext 2435, df-clab 2443 and df-cleq 2449 (and df-sb 1741 and df-v 3111) from vtocl 3161. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theorembj-vtoclg1f1 34604* The FOL content of vtoclg1f 3166 (hence not using ax-ext 2435, df-cleq 2449, df-nfc 2607, df-v 3111). Note the weakened "major" hypothesis and the dv condition between  x and  A (needed since the class-form non-free predicate is not available without ax-ext 2435; as a byproduct, this dispenses with ax-11 1843 and ax-13 2000). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( E. y  y  =  A  ->  ps )
 
Theorembj-vtoclg1f 34605* Reprove vtoclg1f 3166 from bj-vtoclg1f1 34604. This removes dependency on ax-ext 2435, df-cleq 2449 and df-v 3111. Use bj-vtoclg1fv 34606 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theorembj-vtoclg1fv 34606* Version of bj-vtoclg1f 34605 with a dv condition on  x ,  V. This removes dependency on df-sb 1741 and df-clab 2443. Prefer its use over bj-vtoclg1f 34605 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theorembj-rabbida2 34607 Version of rabbidva2 3099 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch )
 ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch } )
 
Theorembj-rabbida 34608 Version of rabbidva 3100 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theorembj-rabbid 34609 Version of rabbidv 3101 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theorembj-rabeqd 34610 Deduction form of rabeq 3103. Note that contrary to rabeq 3103 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ps }
 )
 
Theorembj-rabeqbid 34611 Version of rabeqbidv 3104 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theorembj-rabeqbida 34612 Version of rabeqbidva 3105 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theorembj-seex 34613* Version of seex 4851 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
 |-  F/_ x B   =>    |-  ( ( R Se  A  /\  B  e.  A ) 
 ->  { x  e.  A  |  x R B }  e.  _V )
 
Theorembj-nfcf 34614* Version of df-nfc 2607 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
 |-  F/_ y A   =>    |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
 
Theorembj-axsep2 34615* Remove dependency on ax-13 2000, ax-ext 2435, df-cleq 2449 and df-clel 2452 from axsep2 4579 while shortening its proof. Remark: the comment in zfauscl 4580 is misleading: the essential use of ax-ext 2435 is the one via eleq2 2530 and not the one via vtocl 3161, since the latter can be proved without ax-ext 2435 (see bj-vtocl 34603). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
 )
 
21.29.5.7  Class abstractions

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

 
Theorembj-unrab 34616* Generalization of unrab 3776. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
 |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ps } )  C_  { x  e.  ( A  u.  B )  |  ( ph  \/  ps ) }
 
Theorembj-inrab 34617 Generalization of inrab 3777. (Contributed by BJ, 21-Apr-2019.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  B  |  ps } )  =  { x  e.  ( A  i^i  B )  |  (
 ph  /\  ps ) }
 
Theorembj-inrab2 34618 Shorter proof of inrab 3777. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  /\  ps ) }
 
Theorembj-rabtr 34619* Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
 |-  { x  e.  A  | T.  }  =  A
 
Theorembj-rabtrALT 34620* Alternate proof of bj-rabtr 34619. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  { x  e.  A  | T.  }  =  A
 
Theorembj-rabtrAUTO 34621* Proof of bj-rabtr 34619 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.)
 |-  { x  e.  A  | T.  }  =  A
 
21.29.5.8  Russell's paradox

A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 34622) and then two versions (bj-ru1 34623 and bj-ru 34624). Special attention is put on minimizing axiom depencencies.

 
Theorembj-ru0 34622* The FOL part of Russell's paradox ru 3326 (see also bj-ru1 34623, bj-ru 34624). Use of elequ1 1822, bj-elequ12 34361, bj-spvv 34406 (instead of eleq1 2529, eleq12d 2539, spv 2012 as in ru 3326) permits to remove dependency on ax-11 1843, ax-13 2000, ax-ext 2435, df-sb 1741, df-clab 2443, df-cleq 2449, df-clel 2452. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  -.  A. x ( x  e.  y  <->  -.  x  e.  x )
 
Theorembj-ru1 34623* A version of Russell's paradox ru 3326 (see also bj-ru 34624). Note the more economical use of bj-abeq2 34481 instead of abeq2 2581 to avoid dependency on ax-13 2000. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  -.  E. y  y  =  { x  |  -.  x  e.  x }
 
Theorembj-ru 34624 Remove dependency on ax-13 2000 (and df-v 3111) from Russell's paradox ru 3326 expressed with primitive symbols and with a class variable  V. Note the more economical use of bj-elissetv 34559 instead of isset 3113 to avoid use of df-v 3111. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  -.  { x  |  -.  x  e.  x }  e.  V
 
21.29.5.9  Some disjointness results

A few utility theorems on disjointness of classes.

 
Theorembj-n0i 34625* Inference associated with n0 3803. (Minimizes three statements by a total of 29 bytes.) (Contributed by BJ, 22-Apr-2019.)
 |-  A  =/= 
 (/)   =>    |- 
 E. x  x  e.  A
 
Theorembj-nel0 34626* From the general negation of membership in  A, infer that  A is the empty set. [Could shorten 0xp 5089?] (Contributed by BJ, 6-Oct-2018.)
 |-  -.  x  e.  A   =>    |-  A  =  (/)
 
Theorembj-disjcsn 34627 A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 33914. (Contributed by BJ, 4-Apr-2019.)
 |-  ( A  i^i  { A }
 )  =  (/)
 
Theorembj-disjsn01 34628 Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 34627 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
 |-  ( { (/) }  i^i  { 1o } )  =  (/)
 
Theorembj-1ex 34629  1o is a set. (Contributed by BJ, 6-Apr-2019.)
 |-  1o  e.  _V
 
Theorembj-2ex 34630  2o is a set. (Contributed by BJ, 6-Apr-2019.)
 |-  2o  e.  _V
 
Theorembj-0nel1 34631 The empty set does not belong to 
{ 1o }. (Contributed by BJ, 6-Apr-2019.)
 |-  (/)  e/  { 1o }
 
Theorembj-1nel0 34632  1o does not belong to  { (/) }. (Contributed by BJ, 6-Apr-2019.)
 |-  1o  e/ 
 { (/) }
 
Theorembj-disjdif 34633 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
 |-  (
 ( A  i^i  B )  =  (/)  ->  ( A  \  B )  =  A )
 
21.29.5.10  Complements on direct products

A few utility theorems on direct products.

 
Theorembj-xpimasn 34634 The image of a singleton, general case. [Change and relabel xpimasn 5459 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
 |-  (
 ( A  X.  B ) " { X }
 )  =  if ( X  e.  A ,  B ,  (/) )
 
Theorembj-xpima1sn 34635 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5459 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
 |-  ( X  e/  A  ->  (
 ( A  X.  B ) " { X }
 )  =  (/) )
 
Theorembj-xpima1snALT 34636 Alternate proof of bj-xpima1sn 34635. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( X  e/  A  ->  (
 ( A  X.  B ) " { X }
 )  =  (/) )
 
Theorembj-xpima2sn 34637 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5459] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
 |-  ( X  e.  A  ->  ( ( A  X.  B ) " { X }
 )  =  B )
 
Theorembj-xpnzex 34638 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 exported form (curried form) of xpexcnv 6741 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
 |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )
 
Theorembj-xpexg2 34639 Exported form (curried form) of xpexg 6601. (Contributed by BJ, 2-Apr-2019.)
 |-  ( A  e.  V  ->  ( B  e.  W  ->  ( A  X.  B )  e.  _V ) )
 
Theorembj-xpnzexb 34640 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  e.  ( V  \  { (/) } )  ->  ( B  e.  _V  <->  ( A  X.  B )  e. 
 _V ) )
 
Theorembj-cleq 34641* Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
 |-  ( A  =  B  ->  { x  |  { x }  e.  ( A " C ) }  =  { x  |  { x }  e.  ( B " C ) } )
 
21.29.5.11  "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 34642* If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
 |-  ( A  e.  V  ->  E. x  A  e.  x )
 
Theorembj-snsetex 34643* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 4568. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  ->  { x  |  { x }  e.  A }  e.  _V )
 
Theorembj-clex 34644* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
 |-  ( A  e.  V  ->  { x  |  { x }  e.  ( A " B ) }  e.  _V )
 
Syntaxbj-csngl 34645 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
 class sngl  A
 
Definitiondf-bj-sngl 34646* 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  |  E. y  e.  A  x  =  {
 y } }
 
Theorembj-sngleq 34647 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  -> sngl  A  = sngl  B )
 
Theorembj-elsngl 34648* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
 
Theorembj-snglc 34649 Characterization of the elements of 
A in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  B  <->  { A }  e. sngl  B )
 
Theorembj-snglss 34650 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
 |- sngl  A  C_  ~P A
 
Theorembj-0nelsngl 34651 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7148). (Contributed by BJ, 6-Oct-2018.)
 |-  (/)  e/ sngl  A
 
Theorembj-snglinv 34652* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
 |-  A  =  { x  |  { x }  e. sngl  A }
 
Theorembj-snglex 34653 A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  _V  <-> sngl  A  e.  _V )
 
Syntaxbj-ctag 34654 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
 class tag  A
 
Definitiondf-bj-tag 34655 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  u.  { (/) } )
 
Theorembj-tageq 34656 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  -> tag  A  = tag  B )
 
Theorembj-eltag 34657* Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e. tag  B  <->  ( E. x  e.  B  A  =  { x }  \/  A  =  (/) ) )
 
Theorembj-0eltag 34658 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
 |-  (/)  e. tag  A
 
Theorembj-tagn0 34659 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
 |- tag  A  =/=  (/)
 
Theorembj-tagss 34660 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
 |- tag  A  C_  ~P A
 
Theorembj-snglsstag 34661 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
 |- sngl  A  C_ tag  A
 
Theorembj-sngltagi 34662 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e. sngl  B  ->  A  e. tag  B )
 
Theorembj-sngltag 34663 The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  ->  ( { A }  e. sngl  B  <->  { A }  e. tag  B ) )
 
Theorembj-tagci 34664 Characterization of the elements of 
B in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  B  ->  { A }  e. tag  B )
 
Theorembj-tagcg 34665 Characterization of the elements of 
B in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  e. tag  B ) )
 
Theorembj-taginv 34666* Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
 |-  A  =  { x  |  { x }  e. tag  A }
 
Theorembj-tagex 34667 A class is a set if and only if its tagging is a set. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  _V  <-> tag  A  e.  _V )
 
Theorembj-xtageq 34668 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  X. tag  A )  =  ( C  X. tag  B ) )
 
Theorembj-xtagex 34669 The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)
 |-  ( A  e.  V  ->  ( B  e.  W  ->  ( A  X. tag  B )  e.  _V ) )
 
21.29.5.12  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 34701 and bj-2uplex 34702 (but more importantly, bj-pr21val 34693 and bj-pr22val 34699). 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 4039) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 8052) 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 5056, but here we use "tagged versions" of the factors (see df-bj-tag 34655) 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 34670 Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
 class  ( A Proj 
 B )
 
Definitiondf-bj-proj 34671* 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 }  e.  ( B " { A } ) }
 
Theorembj-projeq 34672 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
 |-  ( A  =  C  ->  ( B  =  D  ->  ( A Proj  B )  =  ( C Proj  D ) ) )
 
Theorembj-projeq2 34673 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
 |-  ( B  =  C  ->  ( A Proj  B )  =  ( A Proj  C ) )
 
Theorembj-projun 34674 The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
 |-  ( A Proj  ( B  u.  C ) )  =  (
 ( A Proj  B )  u.  ( A Proj  C ) )
 
Theorembj-projex 34675 Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
 |-  ( B  e.  V  ->  ( A Proj  B )  e. 
 _V )
 
Theorembj-projval 34676 Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 34291). (Contributed by BJ, 6-Apr-2019.)
 |-  ( A  e.  V  ->  ( A Proj  ( { B }  X. tag  C ) )  =  if ( B  =  A ,  C ,  (/) ) )
 
Syntaxbj-c1upl 34677 Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
 class (| A|)
 
Definitiondf-bj-1upl 34678 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 34692, bj-2uplth 34701, bj-2uplex 34702, and the properties of the projections (see df-bj-pr1 34681 and df-bj-pr2 34695). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
 |- (| A|)  =  ( { (/) }  X. tag  A )
 
Theorembj-1upleq 34679 Substitution property for (|  - |). (Contributed by BJ, 6-Apr-2019.)
 |-  ( A  =  B  -> (| A|)  = (| B|) )
 
Syntaxbj-cpr1 34680 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
 class pr1  A
 
Definitiondf-bj-pr1 34681 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 34682, bj-pr11val 34685, bj-pr21val 34693, bj-pr1ex 34686. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
 |- pr1  A  =  ( (/) Proj  A )
 
Theorembj-pr1eq 34682 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
 |-  ( A  =  B  -> pr1  A  = pr1  B )
 
Theorembj-pr1un 34683 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
 |- pr1  ( A  u.  B )  =  (pr1  A  u. pr1  B )
 
Theorembj-pr1val 34684 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
 |- pr1  ( { A }  X. tag  B )  =  if ( A  =  (/) ,  B ,  (/) )
 
Theorembj-pr11val 34685 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
 |- pr1 (| A|)  =  A
 
Theorembj-pr1ex 34686 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  -> pr1  A  e.  _V )
 
Theorembj-1uplth 34687 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
 |-  ((| A|)  = (| B|)  <->  A  =  B )
 
Theorembj-1uplex 34688 A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
 |-  ((| A|)  e.  _V  <->  A  e.  _V )
 
Theorembj-1upln0 34689 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
 |- (| A|)  =/= 
 (/)
 
Syntaxbj-c2uple 34690 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
 class (| A,  B|)
 
Definitiondf-bj-2upl 34691 Definition of the Morse couple. See df-bj-1upl 34678. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 34692, bj-2uplth 34701, bj-2uplex 34702, and the properties of the projections (see df-bj-pr1 34681 and df-bj-pr2 34695). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |- (| A,  B|)  =  ((| A|)  u.  ( { 1o }  X. tag  B ) )
 
Theorembj-2upleq 34692 Substitution property for (|  - ,  - |). (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  ->  ( C  =  D  -> (| A,  C|)  = (| B,  D|) ) )
 
Theorembj-pr21val 34693 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
 |- pr1 (| A,  B|)  =  A
 
Syntaxbj-cpr2 34694 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
 class pr2  A
 
Definitiondf-bj-pr2 34695 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 34696, bj-pr22val 34699, bj-pr2ex 34700. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |- pr2  A  =  ( 1o Proj  A )
 
Theorembj-pr2eq 34696 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  -> pr2  A  = pr2  B )
 
Theorembj-pr2un 34697 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
 |- pr2  ( A  u.  B )  =  (pr2  A  u. pr2  B )
 
Theorembj-pr2val 34698 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
 |- pr2  ( { A }  X. tag  B )  =  if ( A  =  1o ,  B ,  (/) )
 
Theorembj-pr22val 34699 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
 |- pr2 (| A,  B|)  =  B
 
Theorembj-pr2ex 34700 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  -> pr2  A  e.  _V )
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