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Theorem carden2bex 7011
Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
carden2bex  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem carden2bex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 enen2 6701 . . . . 5  |-  ( A 
~~  B  ->  (
y  ~~  A  <->  y  ~~  B ) )
21rabbidv 2647 . . . 4  |-  ( A 
~~  B  ->  { y  e.  On  |  y 
~~  A }  =  { y  e.  On  |  y  ~~  B }
)
32inteqd 3744 . . 3  |-  ( A 
~~  B  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
43adantr 272 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
5 cardval3ex 7007 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
65adantl 273 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
7 entr 6644 . . . . . 6  |-  ( ( x  ~~  A  /\  A  ~~  B )  ->  x  ~~  B )
87expcom 115 . . . . 5  |-  ( A 
~~  B  ->  (
x  ~~  A  ->  x 
~~  B ) )
98reximdv 2508 . . . 4  |-  ( A 
~~  B  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  On  x  ~~  B
) )
109imp 123 . . 3  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  E. x  e.  On  x  ~~  B
)
11 cardval3ex 7007 . . 3  |-  ( E. x  e.  On  x  ~~  B  ->  ( card `  B )  =  |^| { y  e.  On  | 
y  ~~  B }
)
1210, 11syl 14 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  B )  = 
|^| { y  e.  On  |  y  ~~  B }
)
134, 6, 123eqtr4d 2158 1  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wrex 2392   {crab 2395   |^|cint 3739   class class class wbr 3897   Oncon0 4253   ` cfv 5091    ~~ 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-13 1474  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  ax-un 4323
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-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-er 6395  df-en 6601  df-card 7002
This theorem is referenced by: (None)
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