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Theorem isnumi 6510
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 3796 . . . . 5  |-  ( y  =  A  ->  (
y  ~~  B  <->  A  ~~  B ) )
21rspcev 2702 . . . 4  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. y  e.  On  y  ~~  B )
3 intexrabim 3936 . . . 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 6293 . . . . . 6  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
65simprd 112 . . . . 5  |-  ( A 
~~  B  ->  B  e.  _V )
7 breq2 3797 . . . . . . . . 9  |-  ( x  =  B  ->  (
y  ~~  x  <->  y  ~~  B ) )
87rabbidv 2594 . . . . . . . 8  |-  ( x  =  B  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  B }
)
98inteqd 3649 . . . . . . 7  |-  ( x  =  B  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  B }
)
109eleq1d 2148 . . . . . 6  |-  ( x  =  B  ->  ( |^| { y  e.  On  |  y  ~~  x }  e.  _V  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
1110elrab3 2751 . . . . 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 6508 . . 3  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
1615dmmpt 4846 . 2  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
1714, 16syl6eleqr 2173 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 1285    e. wcel 1434   E.wrex 2350   {crab 2353   _Vcvv 2602   |^|cint 3644   class class class wbr 3793   Oncon0 4126   dom cdm 4371    ~~ cen 6285   cardccrd 6507
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-en 6288  df-card 6508
This theorem is referenced by:  finnum  6511  onenon  6512
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