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Theorem isnumi 6810
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 3848 . . . . 5  |-  ( y  =  A  ->  (
y  ~~  B  <->  A  ~~  B ) )
21rspcev 2722 . . . 4  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. y  e.  On  y  ~~  B )
3 intexrabim 3989 . . . 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 6463 . . . . . 6  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
65simprd 112 . . . . 5  |-  ( A 
~~  B  ->  B  e.  _V )
7 breq2 3849 . . . . . . . . 9  |-  ( x  =  B  ->  (
y  ~~  x  <->  y  ~~  B ) )
87rabbidv 2608 . . . . . . . 8  |-  ( x  =  B  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  B }
)
98inteqd 3693 . . . . . . 7  |-  ( x  =  B  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  B }
)
109eleq1d 2156 . . . . . 6  |-  ( x  =  B  ->  ( |^| { y  e.  On  |  y  ~~  x }  e.  _V  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
1110elrab3 2772 . . . . 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 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 ) )
144, 13mpbird 165 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  { x  e.  _V  |  |^| { y  e.  On  |  y 
~~  x }  e.  _V } )
15 df-card 6808 . . 3  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
1615dmmpt 4926 . 2  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
1714, 16syl6eleqr 2181 1  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   {crab 2363   _Vcvv 2619   |^|cint 3688   class class class wbr 3845   Oncon0 4190   dom cdm 4438    ~~ cen 6455   cardccrd 6807
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-en 6458  df-card 6808
This theorem is referenced by:  finnum  6811  onenon  6812
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