P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ixpsnval 6801*	The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)
|- ( X e. V -> X_ x e. { X } B = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) } )
Theorem	elixp2 6802*	Membership in an infinite Cartesian product. See df-ixp 6799 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
|- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
Theorem	fvixp 6803*	Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
|- ( x = C -> B = D )   =>    |- ( ( F e. X_ x e. A B /\ C e. A ) -> ( F ` C ) e. D )
Theorem	ixpfn 6804*	A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
|- ( F e. X_ x e. A B -> F Fn A )
Theorem	elixp 6805*	Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
|- F e. _V   =>    |- ( F e. X_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
Theorem	elixpconst 6806*	Membership in an infinite Cartesian product of a constant B. (Contributed by NM, 12-Apr-2008.)
|- F e. _V   =>    |- ( F e. X_ x e. A B <-> F : A --> B )
Theorem	ixpconstg 6807*	Infinite Cartesian product of a constant B. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- ( ( A e. V /\ B e. W ) -> X_ x e. A B = ( B ^m A ) )
Theorem	ixpconst 6808*	Infinite Cartesian product of a constant B. (Contributed by NM, 28-Sep-2006.)
|- A e. _V   &    |- B e. _V   =>    |- X_ x e. A B = ( B ^m A )
Theorem	ixpeq1 6809*	Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
|- ( A = B -> X_ x e. A C = X_ x e. B C )
Theorem	ixpeq1d 6810*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
|- ( ph -> A = B )   =>    |- ( ph -> X_ x e. A C = X_ x e. B C )
Theorem	ss2ixp 6811	Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
|- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
Theorem	ixpeq2 6812	Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
|- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
Theorem	ixpeq2dva 6813*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> X_ x e. A B = X_ x e. A C )
Theorem	ixpeq2dv 6814*	Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
|- ( ph -> B = C )   =>    |- ( ph -> X_ x e. A B = X_ x e. A C )
Theorem	cbvixp 6815*	Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- X_ x e. A B = X_ y e. A C
Theorem	cbvixpv 6816*	Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( x = y -> B = C )   =>    |- X_ x e. A B = X_ y e. A C
Theorem	nfixpxy 6817*	Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y X_ x e. A B
Theorem	nfixp1 6818	The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x X_ x e. A B
Theorem	ixpprc 6819*	A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain A, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
|- ( -. A e. _V -> X_ x e. A B = (/) )
Theorem	ixpf 6820*	A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
|- ( F e. X_ x e. A B -> F : A --> U_ x e. A B )
Theorem	uniixp 6821*	The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- U. X_ x e. A B C_ ( A X. U_ x e. A B )
Theorem	ixpexgg 6822*	The existence of an infinite Cartesian product. x is normally a free-variable parameter in B. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( ( A e. W /\ A. x e. A B e. V ) -> X_ x e. A B e. _V )
Theorem	ixpin 6823*	The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- X_ x e. A ( B i^i C ) = ( X_ x e. A B i^i X_ x e. A C )
Theorem	ixpiinm 6824*	The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( E. z z e. B -> X_ x e. A |^|_ y e. B C = |^|_ y e. B X_ x e. A C )
Theorem	ixpintm 6825*	The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( E. z z e. B -> X_ x e. A |^| B = |^|_ y e. B X_ x e. A y )
Theorem	ixp0x 6826	An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
|- X_ x e. (/) A = { (/) }
Theorem	ixpssmap2g 6827*	An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6828 avoids ax-coll 4167. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- ( U_ x e. A B e. V -> X_ x e. A B C_ ( U_ x e. A B ^m A ) )
Theorem	ixpssmapg 6828*	An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- ( A. x e. A B e. V -> X_ x e. A B C_ ( U_ x e. A B ^m A ) )
Theorem	0elixp 6829	Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
|- (/) e. X_ x e. (/) A
Theorem	ixpm 6830*	If an infinite Cartesian product of a family B ( x ) is inhabited, every B ( x ) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
Theorem	ixp0 6831	The infinite Cartesian product of a family B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. x e. A B = (/) -> X_ x e. A B = (/) )
Theorem	ixpssmap 6832*	An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
|- B e. _V   =>    |- X_ x e. A B C_ ( U_ x e. A B ^m A )
Theorem	resixp 6833*	Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
|- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )
Theorem	mptelixpg 6834*	Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
|- ( I e. V -> ( ( x e. I |-> J ) e. X_ x e. I K <-> A. x e. I J e. K ) )
Theorem	elixpsn 6835*	Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( A e. V -> ( F e. X_ x e. { A } B <-> E. y e. B F = { <. A , y >. } ) )
Theorem	ixpsnf1o 6836*	A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. A |-> ( { I } X. { x } ) )   =>    |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )
Theorem	mapsnf1o 6837*	A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. A |-> ( { I } X. { x } ) )   =>    |- ( ( A e. V /\ I e. W ) -> F : A -1-1-onto-> ( A ^m { I } ) )
2.6.28  Equinumerosity
Syntax	cen 6838	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ~~
Syntax	cdom 6839	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ~<_
Syntax	cfn 6840	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 6841*	Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ~~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6848. (Contributed by NM, 28-Mar-1998.)
|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
Definition	df-dom 6842*	Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6852 and domen 6853. (Contributed by NM, 28-Mar-1998.)
|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
Definition	df-fin 6843*	Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our " a e. Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 16050. (Contributed by NM, 22-Aug-2008.)
|- Fin = { x | E. y e. om x ~~ y }
Theorem	relen 6844	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~~
Theorem	reldom 6845	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~<_
Theorem	encv 6846	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
|- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
Theorem	breng 6847*	Equinumerosity relation. This variation of bren 6848 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6848. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
Theorem	bren 6848*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
Theorem	brdom2g 6849*	Dominance relation. This variation of brdomg 6850 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6850. (Revised by BTernaryTau, 29-Nov-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomg 6850*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomi 6851*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~<_ B -> E. f f : A -1-1-> B )
Theorem	brdom 6852*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. f f : A -1-1-> B )
Theorem	domen 6853*	Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) )
Theorem	domeng 6854*	Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
|- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
Theorem	ctex 6855	A class dominated by om is a set. See also ctfoex 7235 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
|- ( A ~<_ om -> A e. _V )
Theorem	f1oen4g 6856	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6861 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1dom4g 6857	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 6862 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )
Theorem	f1oen3g 6858	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6861 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1oen2g 6859	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6861 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1dom2g 6860	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6862 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )
Theorem	f1oeng 6861	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
|- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1domg 6862	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
|- ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) )
Theorem	f1oen 6863	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
|- A e. _V   =>    |- ( F : A -1-1-onto-> B -> A ~~ B )
Theorem	f1dom 6864	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
|- B e. _V   =>    |- ( F : A -1-1-> B -> A ~<_ B )
Theorem	isfi 6865*	Express " A is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
|- ( A e. Fin <-> E. x e. om A ~~ x )
Theorem	enssdom 6866	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
|- ~~ C_ ~<_
Theorem	endom 6867	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
|- ( A ~~ B -> A ~<_ B )
Theorem	enrefg 6868	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A e. V -> A ~~ A )
Theorem	enref 6869	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A e. _V   =>    |- A ~~ A
Theorem	eqeng 6870	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
|- ( A e. V -> ( A = B -> A ~~ B ) )
Theorem	domrefg 6871	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
|- ( A e. V -> A ~<_ A )
Theorem	en2d 6872*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ph -> ( x e. A -> C e. _V ) )   &    |- ( ph -> ( y e. B -> D e. _V ) )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> A ~~ B )
Theorem	en3d 6873*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( y e. B -> D e. A ) )   &    |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )   =>    |- ( ph -> A ~~ B )
Theorem	en2i 6874*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( x e. A -> C e. _V )   &    |- ( y e. B -> D e. _V )   &    |- ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) )   =>    |- A ~~ B
Theorem	en3i 6875*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( x e. A -> C e. B )   &    |- ( y e. B -> D e. A )   &    |- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) )   =>    |- A ~~ B
Theorem	dom2lem 6876*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   =>    |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
Theorem	dom2d 6877*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   =>    |- ( ph -> ( B e. R -> A ~<_ B ) )
Theorem	dom3d 6878*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> A ~<_ B )
Theorem	dom2 6879*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
|- ( x e. A -> C e. B )   &    |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )   =>    |- ( B e. V -> A ~<_ B )
Theorem	dom3 6880*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
|- ( x e. A -> C e. B )   &    |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )   =>    |- ( ( A e. V /\ B e. W ) -> A ~<_ B )
Theorem	idssen 6881	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- _I C_ ~~
Theorem	domssr 6882	If C is a superset of B and B dominates A, then C also dominates A. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )
Theorem	ssdomg 6883	A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( B e. V -> ( A C_ B -> A ~<_ B ) )
Theorem	ener 6884	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ~~ Er _V
Theorem	ensymb 6885	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B <-> B ~~ A )
Theorem	ensym 6886	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B -> B ~~ A )
Theorem	ensymi 6887	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   =>    |- B ~~ A
Theorem	ensymd 6888	Symmetry of equinumerosity. Deduction form of ensym 6886. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A ~~ B )   =>    |- ( ph -> B ~~ A )
Theorem	entr 6889	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
|- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C )
Theorem	domtr 6890	Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	entri 6891	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- A ~~ C
Theorem	entr2i 6892	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- C ~~ A
Theorem	entr3i 6893	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- A ~~ C   =>    |- B ~~ C
Theorem	entr4i 6894	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- C ~~ B   =>    |- A ~~ C
Theorem	endomtr 6895	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	domentr 6896	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C )
Theorem	f1imaeng 6897	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( F : A -1-1-> B /\ C C_ A /\ C e. V ) -> ( F " C ) ~~ C )
Theorem	f1imaen2g 6898	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6899 does not need ax-setind 4593.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
|- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )
Theorem	f1imaen 6899	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
|- C e. _V   =>    |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C )
Theorem	en0 6900	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
|- ( A ~~ (/) <-> A = (/) )