P. 67 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	en1eqsnbi 6601	A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
|- ( A e. B -> ( B ~~ 1o <-> B = { A } ) )
Theorem	snexxph 6602*	A case where the antecedent of snexg 3992 is not needed. The class { x | ph } is from dcextest 4368. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
2.6.30  Schroeder-Bernstein Theorem
Theorem	sbthlem1 6603*	Lemma for isbth 6613. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
Theorem	sbthlem2 6604*	Lemma for isbth 6613. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
Theorem	sbthlemi3 6605*	Lemma for isbth 6613. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
Theorem	sbthlemi4 6606*	Lemma for isbth 6613. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
Theorem	sbthlemi5 6607*	Lemma for isbth 6613. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ ( dom f = A /\ ran g C_ A ) ) -> dom H = A )
Theorem	sbthlemi6 6608*	Lemma for isbth 6613. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
Theorem	sbthlem7 6609*	Lemma for isbth 6613. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( Fun f /\ Fun `' g ) -> Fun H )
Theorem	sbthlemi8 6610*	Lemma for isbth 6613. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
Theorem	sbthlemi9 6611*	Lemma for isbth 6613. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
Theorem	sbthlemi10 6612*	Lemma for isbth 6613. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   &    |- B e. _V   =>    |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Theorem	isbth 6613	Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set A is smaller (has lower cardinality) than B and vice-versa, then A and B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 6603 through sbthlemi10 6612; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6612. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 11343. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.31  Supremum and infimum
Syntax	csup 6614	Extend class notation to include supremum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class sup ( A , B , R )
Syntax	cinf 6615	Extend class notation to include infimum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class inf ( A , B , R )
Definition	df-sup 6616*	Define the supremum of class A. It is meaningful when R is a relation that strictly orders B and when the supremum exists. (Contributed by NM, 22-May-1999.)
|- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\ A. y e. B ( y R x -> E. z e. A y R z ) ) }
Definition	df-inf 6617	Define the infimum of class A. It is meaningful when R is a relation that strictly orders B and when the infimum exists. For example, R could be 'less than', B could be the set of real numbers, and A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
|- inf ( A , B , R ) = sup ( A , B , `' R )
Theorem	supeq1 6618	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
|- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1d 6619	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> B = C )   =>    |- ( ph -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1i 6620	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- B = C   =>    |- sup ( B , A , R ) = sup ( C , A , R )
Theorem	supeq2 6621	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) )
Theorem	supeq3 6622	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
|- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) )
Theorem	supeq123d 6623	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> sup ( A , B , C ) = sup ( D , E , F ) )
Theorem	nfsup 6624	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ x sup ( A , B , R )
Theorem	supmoti 6625*	Any class B has at most one supremum in A (where R is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 7502) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> E* x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
Theorem	supeuti 6626*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> E! x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
Theorem	supval2ti 6627*	Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> sup ( B , A , R ) = ( iota_ x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) ) )
Theorem	eqsupti 6628*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. y e. B -. C R y /\ A. y e. A ( y R C -> E. z e. B y R z ) ) -> sup ( B , A , R ) = C ) )
Theorem	eqsuptid 6629*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ y e. B ) -> -. C R y )   &    |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. z e. B y R z )   =>    |- ( ph -> sup ( B , A , R ) = C )
Theorem	supclti 6630*	A supremum belongs to its base class (closure law). See also supubti 6631 and suplubti 6632. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> sup ( B , A , R ) e. A )
Theorem	supubti 6631*	A supremum is an upper bound. See also supclti 6630 and suplubti 6632. This proof demonstrates how to expand an iota-based definition (df-iota 4943) using riotacl2 5576. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) )
Theorem	suplubti 6632*	A supremum is the least upper bound. See also supclti 6630 and supubti 6631. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
Theorem	suplub2ti 6633*	Bidirectional form of suplubti 6632. (Contributed by Jim Kingdon, 17-Jan-2022.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   &    |- ( ph -> R Or A )   &    |- ( ph -> B C_ A )   =>    |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )
Theorem	supelti 6634*	Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. C ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   &    |- ( ph -> C C_ A )   =>    |- ( ph -> sup ( B , A , R ) e. C )
Theorem	sup00 6635	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- sup ( B , (/) , R ) = (/)
Theorem	supmaxti 6636*	The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ph -> C e. B )   &    |- ( ( ph /\ y e. B ) -> -. C R y )   =>    |- ( ph -> sup ( B , A , R ) = C )
Theorem	supsnti 6637*	The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> B e. A )   =>    |- ( ph -> sup ( { B } , A , R ) = B )
Theorem	isotilem 6638*	Lemma for isoti 6639. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( F Isom R , S ( A , B ) -> ( A. x e. B A. y e. B ( x = y <-> ( -. x S y /\ -. y S x ) ) -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) ) )
Theorem	isoti 6639*	An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( F Isom R , S ( A , B ) -> ( A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) <-> A. u e. B A. v e. B ( u = v <-> ( -. u S v /\ -. v S u ) ) ) )
Theorem	supisolem 6640*	Lemma for supisoti 6642. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   =>    |- ( ( ph /\ D e. A ) -> ( ( A. y e. C -. D R y /\ A. y e. A ( y R D -> E. z e. C y R z ) ) <-> ( A. w e. ( F " C ) -. ( F ` D ) S w /\ A. w e. B ( w S ( F ` D ) -> E. v e. ( F " C ) w S v ) ) ) )
Theorem	supisoex 6641*	Lemma for supisoti 6642. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )   =>    |- ( ph -> E. u e. B ( A. w e. ( F " C ) -. u S w /\ A. w e. B ( w S u -> E. v e. ( F " C ) w S v ) ) )
Theorem	supisoti 6642*	Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )   &    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) )
Theorem	infeq1 6643	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1d 6644	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> B = C )   =>    |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1i 6645	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
|- B = C   =>    |- inf ( B , A , R ) = inf ( C , A , R )
Theorem	infeq2 6646	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )
Theorem	infeq3 6647	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )
Theorem	infeq123d 6648	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )
Theorem	nfinf 6649	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ xinf ( A , B , R )
Theorem	cnvinfex 6650*	Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> E. x e. A ( A. y e. B -. x `' R y /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) )
Theorem	cnvti 6651*	If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )
Theorem	eqinfti 6652*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. y e. B -. y R C /\ A. y e. A ( C R y -> E. z e. B z R y ) ) -> inf ( B , A , R ) = C ) )
Theorem	eqinftid 6653*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ y e. B ) -> -. y R C )   &    |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )   =>    |- ( ph -> inf ( B , A , R ) = C )
Theorem	infvalti 6654*	Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> inf ( B , A , R ) = ( iota_ x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) ) )
Theorem	infclti 6655*	An infimum belongs to its base class (closure law). See also inflbti 6656 and infglbti 6657. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> inf ( B , A , R ) e. A )
Theorem	inflbti 6656*	An infimum is a lower bound. See also infclti 6655 and infglbti 6657. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( C e. B -> -. C Rinf ( B , A , R ) ) )
Theorem	infglbti 6657*	An infimum is the greatest lower bound. See also infclti 6655 and inflbti 6656. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) )
Theorem	infnlbti 6658*	A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. z e. B -. z R C ) -> -. inf ( B , A , R ) R C ) )
Theorem	infminti 6659*	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ph -> C e. B )   &    |- ( ( ph /\ y e. B ) -> -. y R C )   =>    |- ( ph -> inf ( B , A , R ) = C )
Theorem	infmoti 6660*	Any class B has at most one infimum in A (where R is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> E* x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
Theorem	infeuti 6661*	An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> E! x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
Theorem	infsnti 6662*	The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> B e. A )   =>    |- ( ph -> inf ( { B } , A , R ) = B )
Theorem	inf00 6663	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 6664*	Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. y R x /\ A. y e. A ( x R y -> E. z e. C z R y ) ) )   &    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> inf ( ( F " C ) , B , S ) = ( F ` inf ( C , A , R ) ) )
2.6.32  Ordinal isomorphism
Theorem	ordiso2 6665	Generalize ordiso 6666 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 6666*	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
|- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )
2.6.33  Disjoint union
2.6.33.1  Disjoint union
Syntax	cdju 6667	Extend class notation to include disjoint union of two classes.
class ( A ⊔ B )
Definition	df-dju 6668	Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of A or B, tagging each one with whether it came from A or B. (Contributed by Jim Kingdon, 20-Jun-2022.)
|- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
Theorem	djueq12 6669	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )
Theorem	djueq1 6670	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( A ⊔ C ) = ( B ⊔ C ) )
Theorem	djueq2 6671	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( C ⊔ A ) = ( C ⊔ B ) )
Theorem	nfdju 6672	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A ⊔ B )
Theorem	djuex 6673	The disjoint union of sets is a set. (Contributed by AV, 28-Jun-2022.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) e. _V )
2.6.33.2  Left and right injections of a disjoint union In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form (inl |` A ) and (inr |` B ).
Syntax	cinl 6674	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 6675	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 6676	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inl = ( x e. _V |-> <. (/) , x >. )
Definition	df-inr 6677	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inr = ( x e. _V |-> <. 1o , x >. )
Theorem	djulclr 6678	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )
Theorem	djurclr 6679	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
Theorem	djulcl 6680	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
Theorem	djurcl 6681	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
Theorem	djuf1olem 6682*	Lemma for djulf1o 6687 and djurf1o 6688. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. A |-> <. X , x >. )   =>    |- F : A -1-1-onto-> ( { X } X. A )
Theorem	djuf1olemr 6683*	Lemma for djulf1or 6685 and djurf1or 6686. Remark: maybe a version of this lemma with F defined on A and no restriction in the conclusion would be more usable. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )
Theorem	djulclb 6684	Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )
Theorem	djulf1or 6685	The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
|- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
Theorem	djurf1or 6686	The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
|- (inr |` A ) : A -1-1-onto-> ( { 1o } X. A )
Theorem	djulf1o 6687	The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
|- inl : _V -1-1-onto-> ( { (/) } X. _V )
Theorem	djurf1o 6688	The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
|- inr : _V -1-1-onto-> ( { 1o } X. _V )
Theorem	inresflem 6689*	Lemma for inlresf1 6690 and inrresf1 6691. (Contributed by BJ, 4-Jul-2022.)
|- F : A -1-1-onto-> ( { X } X. A )   &    |- ( x e. A -> ( F ` x ) e. B )   =>    |- F : A -1-1-> B
Theorem	inlresf1 6690	The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
|- (inl |` A ) : A -1-1-> ( A ⊔ B )
Theorem	inrresf1 6691	The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
|- (inr |` B ) : B -1-1-> ( A ⊔ B )
Theorem	djuinr 6692	The ranges of any left and right injections are disjoint. Remark: the restrictions seem not necessary, but the proof is not much longer than the proof of |- ( ran inl i^i ran inr ) = (/) (which is easily recovered from it, as in the proof of casefun 6713). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
|- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
Theorem	djuin 6693	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( (inl " A ) i^i (inr " B ) ) = (/)
Theorem	djur 6694*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( C e. ( A ⊔ B ) -> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )
Theorem	djuunr 6695	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
|- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
Theorem	eldju 6696*	Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
|- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
Theorem	djuun 6697	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
|- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2.6.33.3  Universal property of the disjoint union
Theorem	djuss 6698	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
|- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
Theorem	eldju1st 6699	The first component of an element of a disjoint union is either (/) or 1o. (Contributed by AV, 26-Jun-2022.)
|- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
Theorem	eldju2ndl 6700	The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
|- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )