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Theorem carden2bex 7184
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 6837 . . . . 5  |-  ( A 
~~  B  ->  (
y  ~~  A  <->  y  ~~  B ) )
21rabbidv 2726 . . . 4  |-  ( A 
~~  B  ->  { y  e.  On  |  y 
~~  A }  =  { y  e.  On  |  y  ~~  B }
)
32inteqd 3849 . . 3  |-  ( A 
~~  B  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
43adantr 276 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
5 cardval3ex 7180 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
65adantl 277 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
7 entr 6780 . . . . . 6  |-  ( ( x  ~~  A  /\  A  ~~  B )  ->  x  ~~  B )
87expcom 116 . . . . 5  |-  ( A 
~~  B  ->  (
x  ~~  A  ->  x 
~~  B ) )
98reximdv 2578 . . . 4  |-  ( A 
~~  B  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  On  x  ~~  B
) )
109imp 124 . . 3  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  E. x  e.  On  x  ~~  B
)
11 cardval3ex 7180 . . 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 2220 1  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   E.wrex 2456   {crab 2459   |^|cint 3844   class class class wbr 4002   Oncon0 4362   ` cfv 5214    ~~ cen 6734   cardccrd 7174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-er 6531  df-en 6737  df-card 7175
This theorem is referenced by: (None)
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