Theorem isnumi 7055

Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
isnumi	|- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Proof of Theorem isnumi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3940	. . . . 5 |- ( y = A -> ( y ~~ B <-> A ~~ B ) )
2	1	rspcev 2793	. . . 4 |- ( ( A e. On /\ A ~~ B ) -> E. y e. On y ~~ B )
3		intexrabim 4086	. . . 4 |- ( E. y e. On y ~~ B -> |^| { y e. On | y ~~ B } e. _V )
4	2, 3	syl 14	. . 3 |- ( ( A e. On /\ A ~~ B ) -> |^| { y e. On | y ~~ B } e. _V )
5		encv 6648	. . . . . 6 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
6	5	simprd 113	. . . . 5 |- ( A ~~ B -> B e. _V )
7		breq2 3941	. . . . . . . . 9 |- ( x = B -> ( y ~~ x <-> y ~~ B ) )
8	7	rabbidv 2678	. . . . . . . 8 |- ( x = B -> { y e. On | y ~~ x } = { y e. On | y ~~ B } )
9	8	inteqd 3784	. . . . . . 7 |- ( x = B -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ B } )
10	9	eleq1d 2209	. . . . . 6 |- ( x = B -> ( |^| { y e. On | y ~~ x } e. _V <-> |^| { y e. On | y ~~ B } e. _V ) )
11	10	elrab3 2845	. . . . 5 |- ( B e. _V -> ( B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } <-> |^| { y e. On | y ~~ B } e. _V ) )
12	6, 11	syl 14	. . . 4 |- ( A ~~ B -> ( B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } <-> |^| { y e. On | y ~~ B } e. _V ) )
13	12	adantl 275	. . 3 |- ( ( A e. On /\ A ~~ B ) -> ( B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } <-> |^| { y e. On | y ~~ B } e. _V ) )
14	4, 13	mpbird 166	. 2 |- ( ( A e. On /\ A ~~ B ) -> B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } )
15		df-card 7053	. . 3 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
16	15	dmmpt 5042	. 2 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
17	14, 16	eleqtrrdi 2234	1 |- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   E.wrex 2418   {crab 2421   _Vcvv 2689   |^|cint 3779   class class class wbr 3937   Oncon0 4293   dom cdm 4547   ~~ cen 6640   cardccrd 7052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-en 6643  df-card 7053

This theorem is referenced by:  finnum  7056  onenon  7057