Theorem isnumi 7111

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 3968	. . . . 5 |- ( y = A -> ( y ~~ B <-> A ~~ B ) )
2	1	rspcev 2816	. . . 4 |- ( ( A e. On /\ A ~~ B ) -> E. y e. On y ~~ B )
3		intexrabim 4114	. . . 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 6688	. . . . . 6 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
6	5	simprd 113	. . . . 5 |- ( A ~~ B -> B e. _V )
7		breq2 3969	. . . . . . . . 9 |- ( x = B -> ( y ~~ x <-> y ~~ B ) )
8	7	rabbidv 2701	. . . . . . . 8 |- ( x = B -> { y e. On | y ~~ x } = { y e. On | y ~~ B } )
9	8	inteqd 3812	. . . . . . 7 |- ( x = B -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ B } )
10	9	eleq1d 2226	. . . . . 6 |- ( x = B -> ( |^| { y e. On | y ~~ x } e. _V <-> |^| { y e. On | y ~~ B } e. _V ) )
11	10	elrab3 2869	. . . . 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 7109	. . 3 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
16	15	dmmpt 5080	. 2 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
17	14, 16	eleqtrrdi 2251	1 |- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   {crab 2439   _Vcvv 2712   |^|cint 3807   class class class wbr 3965   Oncon0 4323   dom cdm 4585   ~~ cen 6680   cardccrd 7108

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-en 6683  df-card 7109

This theorem is referenced by:  finnum  7112  onenon  7113