Theorem isnumi 6451

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 3788	. . . . 5 |- ( y = A -> ( y ~~ B <-> A ~~ B ) )
2	1	rspcev 2701	. . . 4 |- ( ( A e. On /\ A ~~ B ) -> E. y e. On y ~~ B )
3		intexrabim 3928	. . . 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 6250	. . . . . 6 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
6	5	simprd 112	. . . . 5 |- ( A ~~ B -> B e. _V )
7		breq2 3789	. . . . . . . . 9 |- ( x = B -> ( y ~~ x <-> y ~~ B ) )
8	7	rabbidv 2593	. . . . . . . 8 |- ( x = B -> { y e. On | y ~~ x } = { y e. On | y ~~ B } )
9	8	inteqd 3641	. . . . . . 7 |- ( x = B -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ B } )
10	9	eleq1d 2147	. . . . . 6 |- ( x = B -> ( |^| { y e. On | y ~~ x } e. _V <-> |^| { y e. On | y ~~ B } e. _V ) )
11	10	elrab3 2750	. . . . 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 271	. . 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 165	. 2 |- ( ( A e. On /\ A ~~ B ) -> B e. { x e. _V | |^| { y e. On | y ~~ x } e. _V } )
15		df-card 6449	. . 3 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
16	15	dmmpt 4836	. 2 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
17	14, 16	syl6eleqr 2172	1 |- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   {crab 2352   _Vcvv 2601   |^|cint 3636   class class class wbr 3785   Oncon0 4118   dom cdm 4363   ~~ cen 6242   cardccrd 6448

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-en 6245  df-card 6449

This theorem is referenced by:  finnum  6452  onenon  6453