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Theorem isnumi 7006
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.
StepHypRef Expression
1 breq1 3902 . . . . 5  |-  ( y  =  A  ->  (
y  ~~  B  <->  A  ~~  B ) )
21rspcev 2763 . . . 4  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. y  e.  On  y  ~~  B )
3 intexrabim 4048 . . . 4  |-  ( E. y  e.  On  y  ~~  B  ->  |^| { y  e.  On  |  y 
~~  B }  e.  _V )
42, 3syl 14 . . 3  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  |^| { y  e.  On  |  y  ~~  B }  e.  _V )
5 encv 6608 . . . . . 6  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
65simprd 113 . . . . 5  |-  ( A 
~~  B  ->  B  e.  _V )
7 breq2 3903 . . . . . . . . 9  |-  ( x  =  B  ->  (
y  ~~  x  <->  y  ~~  B ) )
87rabbidv 2649 . . . . . . . 8  |-  ( x  =  B  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  B }
)
98inteqd 3746 . . . . . . 7  |-  ( x  =  B  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  B }
)
109eleq1d 2186 . . . . . 6  |-  ( x  =  B  ->  ( |^| { y  e.  On  |  y  ~~  x }  e.  _V  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
1110elrab3 2814 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
126, 11syl 14 . . . 4  |-  ( A 
~~  B  ->  ( B  e.  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V }  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
1312adantl 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 ) )
144, 13mpbird 166 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V } )
15 df-card 7004 . . 3  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
1615dmmpt 5004 . 2  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
1714, 16eleqtrrdi 2211 1  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394   {crab 2397   _Vcvv 2660   |^|cint 3741   class class class wbr 3899   Oncon0 4255   dom cdm 4509    ~~ cen 6600   cardccrd 7003
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-en 6603  df-card 7004
This theorem is referenced by:  finnum  7007  onenon  7008
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