P. 71 of Theorem List - Metamath Proof Explorer

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elmap 7001	Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^m B ) <-> F : B --> A )
Theorem	mapval2 7002*	Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )
Theorem	elpm 7003	The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^pm B ) <-> ( Fun F /\ F C_ ( B X. A ) ) )
Theorem	elpm2 7004	The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) )
Theorem	fpm 7005	A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F : A --> B -> F e. ( B ^pm A ) )
Theorem	mapsspm 7006	Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- ( A ^m B ) C_ ( A ^pm B )
Theorem	pmsspw 7007	Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( A ^pm B ) C_ ~P ( B X. A )
Theorem	mapsspw 7008	Set exponentiation is a subset of the power set of the cross product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ^m B ) C_ ~P ( B X. A )
Theorem	fvmptmap 7009*	Special case of fvmpt 5765 for operator theorems. (Contributed by NM, 27-Nov-2007.)
|- C e. _V   &    |- D e. _V   &    |- R e. _V   &    |- ( x = A -> B = C )   &    |- F = ( x e. ( R ^m D ) |-> B )   =>    |- ( A : D --> R -> ( F ` A ) = C )
Theorem	map0e 7010	Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( A e. V -> ( A ^m (/) ) = 1o )
Theorem	map0b 7011	Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A =/= (/) -> ( (/) ^m A ) = (/) )
Theorem	map0g 7012	Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )
Theorem	map0 7013	Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) )
Theorem	mapsn 7014*	The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } }
Theorem	mapss 7015	Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( B e. V /\ A C_ B ) -> ( A ^m C ) C_ ( B ^m C ) )
Theorem	fdiagfn 7016*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. B |-> ( I X. { x } ) )   =>    |- ( ( B e. V /\ I e. W ) -> F : B --> ( B ^m I ) )
Theorem	fvdiagfn 7017*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. B |-> ( I X. { x } ) )   =>    |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )
Theorem	mapsnconst 7018	Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   =>    |- ( F e. ( B ^m S ) -> F = ( S X. { ( F ` X ) } ) )
Theorem	mapsncnv 7019*	Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( x e. ( B ^m S ) |-> ( x ` X ) )   =>    |- `' F = ( y e. B |-> ( S X. { y } ) )
Theorem	mapsnf1o2 7020*	Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( x e. ( B ^m S ) |-> ( x ` X ) )   =>    |- F : ( B ^m S ) -1-1-onto-> B
Theorem	mapsnf1o3 7021*	Explicit bijection in the reverse of mapsnf1o2 7020. (Contributed by Stefan O'Rear, 24-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( y e. B |-> ( S X. { y } ) )   =>    |- F : B -1-1-onto-> ( B ^m S )
2.4.30  Infinite Cartesian products
Syntax	cixp 7022	Extend class notation to include infinite Cartesian products.
class X_ x e. A B
Definition	df-ixp 7023*	Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with x e. A written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually B represents a class expression containing x free and thus can be thought of as B ( x ). Normally, x is not free in A, although this is not a requirement of the definition. (Contributed by NM, 28-Sep-2006.)
|- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
Theorem	dfixp 7024*	Eliminate the expression { x | x e. A } in df-ixp 7023, under the assumption that A and x are disjoint. This way, we can say that x is bound in X_ x e. A B even if it appears free in A. (Contributed by Mario Carneiro, 12-Aug-2016.)
|- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
Theorem	elixp2 7025*	Membership in an infinite Cartesian product. See df-ixp 7023 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 7026*	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 7027*	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 7028*	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 7029*	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 7030*	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 7031*	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 7032*	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 7033*	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 7034	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 7035	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 7036*	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 7037*	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 7038*	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 7039*	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	nfixp 7040	Bound-variable hypothesis builder for indexed cross product. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y X_ x e. A B
Theorem	nfixp1 7041	The index variable in an indexed cross 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 7042*	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 7043*	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 7044*	The union of an infinite Cartesian product is included in a cross 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	ixpexg 7045*	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 Mario Carneiro, 25-Jan-2015.)
|- ( A. x e. A B e. V -> X_ x e. A B e. _V )
Theorem	ixpin 7046*	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	ixpiin 7047*	The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
|- ( B =/= (/) -> X_ x e. A |^|_ y e. B C = |^|_ y e. B X_ x e. A C )
Theorem	ixpint 7048*	The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- ( B =/= (/) -> X_ x e. A |^| B = |^|_ y e. B X_ x e. A y )
Theorem	ixp0x 7049	An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
|- X_ x e. (/) A = { (/) }
Theorem	ixpssmap2g 7050*	An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7051 avoids ax-rep 4280. (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 7051*	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 7052	Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
|- (/) e. X_ x e. (/) A
Theorem	ixpn0 7053	The infinite Cartesian product of a family B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8319. (Contributed by Mario Carneiro, 22-Jun-2016.)
|- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) )
Theorem	ixp0 7054	The infinite Cartesian product of a family B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8319. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
|- ( E. x e. A B = (/) -> X_ x e. A B = (/) )
Theorem	ixpssmap 7055*	An infinite Cartesian product is a subset of set exponentation. 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 7056*	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	undifixp 7057*	Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)
|- ( ( F e. X_ x e. B C /\ G e. X_ x e. ( A \ B ) C /\ B C_ A ) -> ( F u. G ) e. X_ x e. A C )
Theorem	mptelixpg 7058*	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	resixpfo 7059*	Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.)
|- F = ( f e. X_ x e. A C |-> ( f |` B ) )   =>    |- ( ( B C_ A /\ X_ x e. A C =/= (/) ) -> F : X_ x e. A C -onto-> X_ x e. B C )
Theorem	elixpsn 7060*	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 7061*	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 7062*	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 } ) )
Theorem	boxriin 7063*	A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A. x e. I A C_ B -> X_ x e. I A = ( X_ x e. I B i^i |^|_ y e. I X_ x e. I if ( x = y , A , B ) ) )
Theorem	boxcutc 7064*	The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( X e. A /\ A. k e. A C C_ B ) -> ( X_ k e. A B \ X_ k e. A if ( k = X , C , B ) ) = X_ k e. A if ( k = X , ( B \ C ) , B ) )
2.4.31  Equinumerosity
Syntax	cen 7065	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ~~
Syntax	cdom 7066	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ~<_
Syntax	csdm 7067	Extend class definition to include the strict dominance relation (curly less-than)
class ~<
Syntax	cfn 7068	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 7069*	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 7076. (Contributed by NM, 28-Mar-1998.)
|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
Definition	df-dom 7070*	Define the dominance relation. For an alternate definition see dfdom2 7092. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 7079 and domen 7080. (Contributed by NM, 28-Mar-1998.)
|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
Definition	df-sdom 7071	Define the strict dominance relation. Alternate possible definitions are derived as brsdom 7089 and brsdom2 7190. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
|- ~< = ( ~<_ \ ~~ )
Definition	df-fin 7072*	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 7552. If we accept Infinity, we can also express A e. Fin by A ~< om (theorem isfinite 7563.) (Contributed by NM, 22-Aug-2008.)
|- Fin = { x | E. y e. om x ~~ y }
Theorem	relen 7073	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~~
Theorem	reldom 7074	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~<_
Theorem	relsdom 7075	Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
|- Rel ~<
Theorem	bren 7076*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
Theorem	brdomg 7077*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomi 7078*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~<_ B -> E. f f : A -1-1-> B )
Theorem	brdom 7079*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. f f : A -1-1-> B )
Theorem	domen 7080*	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 7081*	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	f1oen3g 7082	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7085 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 7083	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7085 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 7084	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7086 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 7085	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 7086	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 7087	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 7088	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	brsdom 7089	Strict dominance relation, meaning " B is strictly greater in size than A." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
|- ( A ~< B <-> ( A ~<_ B /\ -. A ~~ B ) )
Theorem	isfi 7090*	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 7091	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
|- ~~ C_ ~<_
Theorem	dfdom2 7092	Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
|- ~<_ = ( ~< u. ~~ )
Theorem	endom 7093	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
|- ( A ~~ B -> A ~<_ B )
Theorem	sdomdom 7094	Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
|- ( A ~< B -> A ~<_ B )
Theorem	sdomnen 7095	Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
|- ( A ~< B -> -. A ~~ B )
Theorem	brdom2 7096	Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
|- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
Theorem	bren2 7097	Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
|- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )
Theorem	enrefg 7098	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 7099	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A e. _V   =>    |- A ~~ A
Theorem	eqeng 7100	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
|- ( A e. V -> ( A = B -> A ~~ B ) )