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Theorem isnumi 8115
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 variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 4294 . . 3  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
21rspcev 3072 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. x  e.  On  x  ~~  B )
3 isnum2 8114 . 2  |-  ( B  e.  dom  card  <->  E. x  e.  On  x  ~~  B
)
42, 3sylibr 212 1  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   E.wrex 2715   class class class wbr 4291   Oncon0 4718   dom cdm 4839    ~~ cen 7306   cardccrd 8104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5419  df-fn 5420  df-f 5421  df-en 7310  df-card 8108
This theorem is referenced by:  finnum  8117  onenon  8118  tskwe  8119  xpnum  8120  isnum3  8123  dfac8alem  8198  cdanum  8367  fin67  8563  isfin7-2  8564  gch2  8841  gchacg  8846  znnen  13494  qnnen  13495  met1stc  20095  re2ndc  20377  uniiccdif  21057  dyadmbl  21079  opnmblALT  21082  mbfimaopnlem  21132  aannenlem3  21795
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