Theorem isnumi 7159

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 3992	. . . . 5 |- ( y = A -> ( y ~~ B <-> A ~~ B ) )
2	1	rspcev 2834	. . . 4 |- ( ( A e. On /\ A ~~ B ) -> E. y e. On y ~~ B )
3		intexrabim 4139	. . . 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 6724	. . . . . 6 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
6	5	simprd 113	. . . . 5 |- ( A ~~ B -> B e. _V )
7		breq2 3993	. . . . . . . . 9 |- ( x = B -> ( y ~~ x <-> y ~~ B ) )
8	7	rabbidv 2719	. . . . . . . 8 |- ( x = B -> { y e. On | y ~~ x } = { y e. On | y ~~ B } )
9	8	inteqd 3836	. . . . . . 7 |- ( x = B -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ B } )
10	9	eleq1d 2239	. . . . . 6 |- ( x = B -> ( |^| { y e. On | y ~~ x } e. _V <-> |^| { y e. On | y ~~ B } e. _V ) )
11	10	elrab3 2887	. . . . 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 7157	. . 3 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
16	15	dmmpt 5106	. 2 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
17	14, 16	eleqtrrdi 2264	1 |- ( ( A e. On /\ A ~~ B ) -> B e. dom card )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   |^|cint 3831   class class class wbr 3989   Oncon0 4348   dom cdm 4611   ~~ cen 6716   cardccrd 7156

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-en 6719  df-card 7157

This theorem is referenced by:  finnum  7160  onenon  7161