P. 91 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cda0en 9001	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A e. V -> ( A +c (/) ) ~~ A )
Theorem	xp2cda 9002	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A e. V -> ( A X. 2o ) = ( A +c A ) )
Theorem	cdacomen 9003	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A +c B ) ~~ ( B +c A )
Theorem	cdaassen 9004	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A +c B ) +c C ) ~~ ( A +c ( B +c C ) ) )
Theorem	xpcdaen 9005	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B +c C ) ) ~~ ( ( A X. B ) +c ( A X. C ) ) )
Theorem	mapcdaen 9006	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ^m ( B +c C ) ) ~~ ( ( A ^m B ) X. ( A ^m C ) ) )
Theorem	pwcdaen 9007	Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( A e. V /\ B e. W ) -> ~P ( A +c B ) ~~ ( ~P A X. ~P B ) )
Theorem	cdadom1 9008	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A ~<_ B -> ( A +c C ) ~<_ ( B +c C ) )
Theorem	cdadom2 9009	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A ~<_ B -> ( C +c A ) ~<_ ( C +c B ) )
Theorem	cdadom3 9010	A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> A ~<_ ( A +c B ) )
Theorem	cdaxpdom 9011	Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
|- ( ( 1o ~< A /\ 1o ~< B ) -> ( A +c B ) ~<_ ( A X. B ) )
Theorem	cdafi 9012	The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
|- ( ( A ~< om /\ B ~< om ) -> ( A +c B ) ~< om )
Theorem	cdainflem 9013	Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
|- ( ( A u. B ) ~~ om -> ( A ~~ om \/ B ~~ om ) )
Theorem	cdainf 9014	A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( om ~<_ A <-> om ~<_ ( A +c A ) )
Theorem	infcda1 9015	An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( om ~<_ A -> ( A +c 1o ) ~~ A )
Theorem	pwcda1 9016	The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( A e. V -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )
Theorem	pwcdaidm 9017	If the natural numbers inject into A, then ~P A is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( om ~<_ A -> ( ~P A +c ~P A ) ~~ ~P A )
Theorem	cdalepw 9018	If A is idempotent under cardinal sum and B is dominated by the power set of A, then so is the cardinal sum of A and B. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )
Theorem	onacda 9019	The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) ~~ ( A +c B ) )
Theorem	cardacda 9020	The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. dom card /\ B e. dom card ) -> ( A +c B ) ~~ ( ( card ` A ) +o ( card ` B ) ) )
Theorem	cdanum 9021	The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ B e. dom card ) -> ( A +c B ) e. dom card )
Theorem	unnum 9022	The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ B e. dom card ) -> ( A u. B ) e. dom card )
Theorem	nnacda 9023	The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
|- ( ( A e. om /\ B e. om ) -> ( card ` ( A +c B ) ) = ( A +o B ) )
Theorem	ficardun 9024	The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> ( card ` ( A u. B ) ) = ( ( card ` A ) +o ( card ` B ) ) )
Theorem	ficardun2 9025	The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( card ` ( A u. B ) ) C_ ( ( card ` A ) +o ( card ` B ) ) )
Theorem	pwsdompw 9026*	Lemma for domtriom 9265. This is the equinumerosity version of the algebraic identity sum_ k e. n ( 2 ^ k ) = ( 2 ^ n ) - 1. (Contributed by Mario Carneiro, 7-Feb-2013.)
|- ( ( n e. om /\ A. k e. suc n ( B ` k ) ~~ ~P k ) -> U_ k e. n ( B ` k ) ~< ( B ` n ) )
Theorem	unctb 9027	The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
|- ( ( A ~<_ om /\ B ~<_ om ) -> ( A u. B ) ~<_ om )
Theorem	infcdaabs 9028	Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A +c B ) ~~ A )
Theorem	infunabs 9029	An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A u. B ) ~~ A )
Theorem	infcda 9030	The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~~ ( A u. B ) )
Theorem	infdif 9031	The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) ~~ A )
Theorem	infdif2 9032	Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( ( A \ B ) ~<_ B <-> A ~<_ B ) )
Theorem	infxpdom 9033	Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. dom card /\ om ~<_ A /\ B ~<_ A ) -> ( A X. B ) ~<_ A )
Theorem	infxpabs 9034	Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B =/= (/) /\ B ~<_ A ) ) -> ( A X. B ) ~~ A )
Theorem	infunsdom1 9035	The union of two sets that are strictly dominated by the infinite set X is also dominated by X. This version of infunsdom 9036 assumes additionally that A is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
|- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> ( A u. B ) ~< X )
Theorem	infunsdom 9036	The union of two sets that are strictly dominated by the infinite set X is also strictly dominated by X. (Contributed by Mario Carneiro, 3-May-2015.)
|- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~< X /\ B ~< X ) ) -> ( A u. B ) ~< X )
Theorem	infxp 9037	Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( ( A e. dom card /\ om ~<_ A ) /\ ( B e. dom card /\ B =/= (/) ) ) -> ( A X. B ) ~~ ( A u. B ) )
Theorem	pwcdadom 9038	A property of dominance over a powerset, and a main lemma for gchac 9503. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )
Theorem	infpss 9039*	Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9135. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( om ~<_ A -> E. x ( x C. A /\ x ~~ A ) )
Theorem	infmap2 9040*	An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9398 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( ( om ~<_ A /\ B ~<_ A /\ ( A ^m B ) e. dom card ) -> ( A ^m B ) ~~ { x | ( x C_ A /\ x ~~ B ) } )
2.6.10  The Ackermann bijection
Theorem	ackbij2lem1 9041	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( A e. om -> ~P A C_ ( ~P om i^i Fin ) )
Theorem	ackbij1lem1 9042	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( -. A e. B -> ( B i^i suc A ) = ( B i^i A ) )
Theorem	ackbij1lem2 9043	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( A e. B -> ( B i^i suc A ) = ( { A } u. ( B i^i A ) ) )
Theorem	ackbij1lem3 9044	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( A e. om -> A e. ( ~P om i^i Fin ) )
Theorem	ackbij1lem4 9045	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 19-Nov-2014.)
|- ( A e. om -> { A } e. ( ~P om i^i Fin ) )
Theorem	ackbij1lem5 9046	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 19-Nov-2014.)
|- ( A e. om -> ( card ` ~P suc A ) = ( ( card ` ~P A ) +o ( card ` ~P A ) ) )
Theorem	ackbij1lem6 9047	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( ( A e. ( ~P om i^i Fin ) /\ B e. ( ~P om i^i Fin ) ) -> ( A u. B ) e. ( ~P om i^i Fin ) )
Theorem	ackbij1lem7 9048*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 21-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( A e. ( ~P om i^i Fin ) -> ( F ` A ) = ( card ` U_ y e. A ( { y } X. ~P y ) ) )
Theorem	ackbij1lem8 9049*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 19-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( A e. om -> ( F ` { A } ) = ( card ` ~P A ) )
Theorem	ackbij1lem9 9050*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 19-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( ( A e. ( ~P om i^i Fin ) /\ B e. ( ~P om i^i Fin ) /\ ( A i^i B ) = (/) ) -> ( F ` ( A u. B ) ) = ( ( F ` A ) +o ( F ` B ) ) )
Theorem	ackbij1lem10 9051*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- F : ( ~P om i^i Fin ) --> om
Theorem	ackbij1lem11 9052*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( ( A e. ( ~P om i^i Fin ) /\ B C_ A ) -> B e. ( ~P om i^i Fin ) )
Theorem	ackbij1lem12 9053*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( ( B e. ( ~P om i^i Fin ) /\ A C_ B ) -> ( F ` A ) C_ ( F ` B ) )
Theorem	ackbij1lem13 9054*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( F ` (/) ) = (/)
Theorem	ackbij1lem14 9055*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( A e. om -> ( F ` { A } ) = suc ( F ` A ) )
Theorem	ackbij1lem15 9056*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( ( ( A e. ( ~P om i^i Fin ) /\ B e. ( ~P om i^i Fin ) ) /\ ( c e. om /\ c e. A /\ -. c e. B ) ) -> -. ( F ` ( A i^i suc c ) ) = ( F ` ( B i^i suc c ) ) )
Theorem	ackbij1lem16 9057*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( ( A e. ( ~P om i^i Fin ) /\ B e. ( ~P om i^i Fin ) ) -> ( ( F ` A ) = ( F ` B ) -> A = B ) )
Theorem	ackbij1lem17 9058*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- F : ( ~P om i^i Fin ) -1-1-> om
Theorem	ackbij1lem18 9059*	Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( A e. ( ~P om i^i Fin ) -> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc ( F ` A ) )
Theorem	ackbij1 9060*	The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- F : ( ~P om i^i Fin ) -1-1-onto-> om
Theorem	ackbij1b 9061*	The Ackermann bijection, part 1b: the bijection from ackbij1 9060 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   =>    |- ( A e. om -> ( F " ~P A ) = ( card ` ~P A ) )
Theorem	ackbij2lem2 9062*	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   &    |- G = ( x e. _V |-> ( y e. ~P dom x |-> ( F ` ( x " y ) ) ) )   =>    |- ( A e. om -> ( rec ( G , (/) ) ` A ) : ( R1 ` A ) -1-1-onto-> ( card ` ( R1 ` A ) ) )
Theorem	ackbij2lem3 9063*	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   &    |- G = ( x e. _V |-> ( y e. ~P dom x |-> ( F ` ( x " y ) ) ) )   =>    |- ( A e. om -> ( rec ( G , (/) ) ` A ) C_ ( rec ( G , (/) ) ` suc A ) )
Theorem	ackbij2lem4 9064*	Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   &    |- G = ( x e. _V |-> ( y e. ~P dom x |-> ( F ` ( x " y ) ) ) )   =>    |- ( ( ( A e. om /\ B e. om ) /\ B C_ A ) -> ( rec ( G , (/) ) ` B ) C_ ( rec ( G , (/) ) ` A ) )
Theorem	ackbij2 9065*	The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )   &    |- G = ( x e. _V |-> ( y e. ~P dom x |-> ( F ` ( x " y ) ) ) )   &    |- H = U. ( rec ( G , (/) ) " om )   =>    |- H : U. ( R1 " om ) -1-1-onto-> om
Theorem	r1om 9066	The set of hereditarily finite sets is countable. See ackbij2 9065 for an explicit bijection that works without Infinity. See also r1omALT 9598. (Contributed by Stefan O'Rear, 18-Nov-2014.)
|- ( R1 ` om ) ~~ om
Theorem	fictb 9067	A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. B -> ( A ~<_ om <-> ( fi ` A ) ~<_ om ) )
2.6.11  Cofinality (without Axiom of Choice)
Theorem	cflem 9068*	A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set A. (Contributed by NM, 24-Apr-2004.)
|- ( A e. V -> E. x E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) )
Theorem	cfval 9069*	Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A, there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
Theorem	cff 9070	Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- cf : On --> On
Theorem	cfub 9071*	An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( cf ` A ) C_ |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A C_ U. y ) ) }
Theorem	cflm 9072*	Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
|- ( ( A e. B /\ Lim A ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
Theorem	cf0 9073	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
|- ( cf ` (/) ) = (/)
Theorem	cardcf 9074	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( card ` ( cf ` A ) ) = ( cf ` A )
Theorem	cflecard 9075	Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( cf ` A ) C_ ( card ` A )
Theorem	cfle 9076	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( cf ` A ) C_ A
Theorem	cfon 9077	The cofinality of any set is an ordinal (although it only makes sense when A is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
|- ( cf ` A ) e. On
Theorem	cfeq0 9078	Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
|- ( A e. On -> ( ( cf ` A ) = (/) <-> A = (/) ) )
Theorem	cfsuc 9079	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
|- ( A e. On -> ( cf ` suc A ) = 1o )
Theorem	cff1 9080*	There is always a map from ( cf ` A ) to A (this is a stronger condition than the definition, which only presupposes a map from some y ~~ ( cf ` A ). (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ( A e. On -> E. f ( f : ( cf ` A ) -1-1-> A /\ A. z e. A E. w e. ( cf ` A ) z C_ ( f ` w ) ) )
Theorem	cfflb 9081*	If there is a cofinal map from B to A, then B is at least ( cf ` A ). This theorem and cff1 9080 motivate the picture of ( cf ` A ) as the greatest lower bound of the domain of cofinal maps into A. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ( ( A e. On /\ B e. On ) -> ( E. f ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ( cf ` A ) C_ B ) )
Theorem	cfval2 9082*	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ( A e. On -> ( cf ` A ) = |^|_ x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } ( card ` x ) )
Theorem	coflim 9083*	A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ( ( Lim A /\ B C_ A ) -> ( U. B = A <-> A. x e. A E. y e. B x C_ y ) )
Theorem	cflim3 9084*	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- A e. _V   =>    |- ( Lim A -> ( cf ` A ) = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )
Theorem	cflim2 9085	The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- A e. _V   =>    |- ( Lim A <-> Lim ( cf ` A ) )
Theorem	cfom 9086	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
|- ( cf ` om ) = om
Theorem	cfss 9087*	There is a cofinal subset of A of cardinality ( cf ` A ). (Contributed by Mario Carneiro, 24-Jun-2013.)
|- A e. _V   =>    |- ( Lim A -> E. x ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) )
Theorem	cfslb 9088	Any cofinal subset of A is at least as large as ( cf ` A ). (Contributed by Mario Carneiro, 24-Jun-2013.)
|- A e. _V   =>    |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) ~<_ B )
Theorem	cfslbn 9089	Any subset of A smaller than its cofinality has union less than A. (This is the contrapositive to cfslb 9088.) (Contributed by Mario Carneiro, 24-Jun-2013.)
|- A e. _V   =>    |- ( ( Lim A /\ B C_ A /\ B ~< ( cf ` A ) ) -> U. B e. A )
Theorem	cfslb2n 9090*	Any small collection of small subsets of A cannot have union A, where "small" means smaller than the cofinality. This is a stronger version of cfslb 9088. This is a common application of cofinality: under AC, ( aleph ` 1 ) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
|- A e. _V   =>    |- ( ( Lim A /\ A. x e. B ( x C_ A /\ x ~< ( cf ` A ) ) ) -> ( B ~< ( cf ` A ) -> U. B =/= A ) )
Theorem	cofsmo 9091*	Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- C = { y e. B | A. w e. y ( f ` w ) e. ( f ` y ) }   &    |- K = |^| { x e. B | z C_ ( f ` x ) }   &    |- O = OrdIso ( _E , C )   =>    |- ( ( Ord A /\ B e. On ) -> ( E. f ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> E. x e. suc B E. g ( g : x --> A /\ Smo g /\ A. z e. A E. v e. x z C_ ( g ` v ) ) ) )
Theorem	cfsmolem 9092*	Lemma for cfsmo 9093. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- F = ( z e. _V |-> ( ( g ` dom z ) u. U_ t e. dom z suc ( z ` t ) ) )   &    |- G = (recs ( F ) |` ( cf ` A ) )   =>    |- ( A e. On -> E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. z e. A E. w e. ( cf ` A ) z C_ ( f ` w ) ) )
Theorem	cfsmo 9093*	The map in cff1 9080 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ( A e. On -> E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. z e. A E. w e. ( cf ` A ) z C_ ( f ` w ) ) )
Theorem	cfcoflem 9094*	Lemma for cfcof 9096, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
|- ( ( A e. On /\ B e. On ) -> ( E. f ( f : B --> A /\ Smo f /\ A. x e. A E. y e. B x C_ ( f ` y ) ) -> ( cf ` A ) C_ ( cf ` B ) ) )
Theorem	coftr 9095*	If there is a cofinal map from B to A and another from C to A, then there is also a cofinal map from C to B. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 9096. (Contributed by Mario Carneiro, 16-Mar-2013.)
|- H = ( t e. C |-> |^| { n e. B | ( g ` t ) C_ ( f ` n ) } )   =>    |- ( E. f ( f : B --> A /\ Smo f /\ A. x e. A E. y e. B x C_ ( f ` y ) ) -> ( E. g ( g : C --> A /\ A. z e. A E. w e. C z C_ ( g ` w ) ) -> E. h ( h : C --> B /\ A. s e. B E. w e. C s C_ ( h ` w ) ) ) )
Theorem	cfcof 9096*	If there is a cofinal map from A to B, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof ( A , B ) and defines our cf ( B ) as the minimum B such that cof ( A , B ). (Contributed by Mario Carneiro, 20-Mar-2013.)
|- ( ( A e. On /\ B e. On ) -> ( E. f ( f : B --> A /\ Smo f /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ( cf ` A ) = ( cf ` B ) ) )
Theorem	cfidm 9097	The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( cf ` ( cf ` A ) ) = ( cf ` A )
Theorem	alephsing 9098	The cofinality of a limit aleph is the same as the cofinality of its argument, so if ( aleph ` A ) < A, then ( aleph ` A ) is singular. Conversely, if ( aleph ` A ) is regular (i.e. weakly inaccessible), then ( aleph ` A ) = A, so A has to be rather large (see alephfp 8931). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
|- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
2.6.12  Eight inequivalent definitions of finite set
Theorem	sornom 9099*	The range of a single-step monotone function from om into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)
|- ( ( F Fn om /\ A. a e. om ( ( F ` a ) R ( F ` suc a ) \/ ( F ` a ) = ( F ` suc a ) ) /\ R Po ran F ) -> R Or ran F )
Syntax	cfin1a 9100	Extend class notation to include the class of Ia-finite sets.
class FinIa