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Theorem isnumi 7004
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 3900 . . . . 5  |-  ( y  =  A  ->  (
y  ~~  B  <->  A  ~~  B ) )
21rspcev 2761 . . . 4  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. y  e.  On  y  ~~  B )
3 intexrabim 4046 . . . 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 6606 . . . . . 6  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
65simprd 113 . . . . 5  |-  ( A 
~~  B  ->  B  e.  _V )
7 breq2 3901 . . . . . . . . 9  |-  ( x  =  B  ->  (
y  ~~  x  <->  y  ~~  B ) )
87rabbidv 2647 . . . . . . . 8  |-  ( x  =  B  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  B }
)
98inteqd 3744 . . . . . . 7  |-  ( x  =  B  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  B }
)
109eleq1d 2184 . . . . . 6  |-  ( x  =  B  ->  ( |^| { y  e.  On  |  y  ~~  x }  e.  _V  <->  |^| { y  e.  On  |  y  ~~  B }  e.  _V ) )
1110elrab3 2812 . . . . 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 273 . . 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 7002 . . 3  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
1615dmmpt 5002 . 2  |-  dom  card  =  { x  e.  _V  |  |^| { y  e.  On  |  y  ~~  x }  e.  _V }
1714, 16syl6eleqr 2209 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 1314    e. wcel 1463   E.wrex 2392   {crab 2395   _Vcvv 2658   |^|cint 3739   class class class wbr 3897   Oncon0 4253   dom cdm 4507    ~~ cen 6598   cardccrd 7001
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-en 6601  df-card 7002
This theorem is referenced by:  finnum  7005  onenon  7006
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