Theorem isnumi 7183

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 4008	. . . . 5 |- ( y = A -> ( y ~~ B <-> A ~~ B ) )
2	1	rspcev 2843	. . . 4 |- ( ( A e. On /\ A ~~ B ) -> E. y e. On y ~~ B )
3		intexrabim 4155	. . . 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 6748	. . . . . 6 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
6	5	simprd 114	. . . . 5 |- ( A ~~ B -> B e. _V )
7		breq2 4009	. . . . . . . . 9 |- ( x = B -> ( y ~~ x <-> y ~~ B ) )
8	7	rabbidv 2728	. . . . . . . 8 |- ( x = B -> { y e. On | y ~~ x } = { y e. On | y ~~ B } )
9	8	inteqd 3851	. . . . . . 7 |- ( x = B -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ B } )
10	9	eleq1d 2246	. . . . . 6 |- ( x = B -> ( |^| { y e. On | y ~~ x } e. _V <-> |^| { y e. On | y ~~ B } e. _V ) )
11	10	elrab3 2896	. . . . 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 277	. . 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 167	. 2 |- ( ( A e. On /\ A ~~ B ) -> B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } )
15		df-card 7181	. . 3 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
16	15	dmmpt 5126	. 2 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
17	14, 16	eleqtrrdi 2271	1 |- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   _Vcvv 2739   |^|cint 3846   class class class wbr 4005   Oncon0 4365   dom cdm 4628   ~~ cen 6740   cardccrd 7180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-en 6743  df-card 7181

This theorem is referenced by:  finnum  7184  onenon  7185