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Theorem carden2bex 7159
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 6817 . . . . 5  |-  ( A 
~~  B  ->  (
y  ~~  A  <->  y  ~~  B ) )
21rabbidv 2719 . . . 4  |-  ( A 
~~  B  ->  { y  e.  On  |  y 
~~  A }  =  { y  e.  On  |  y  ~~  B }
)
32inteqd 3834 . . 3  |-  ( A 
~~  B  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
43adantr 274 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
5 cardval3ex 7155 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
65adantl 275 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
7 entr 6760 . . . . . 6  |-  ( ( x  ~~  A  /\  A  ~~  B )  ->  x  ~~  B )
87expcom 115 . . . . 5  |-  ( A 
~~  B  ->  (
x  ~~  A  ->  x 
~~  B ) )
98reximdv 2571 . . . 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 7155 . . 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 2213 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 1348   E.wrex 2449   {crab 2452   |^|cint 3829   class class class wbr 3987   Oncon0 4346   ` cfv 5196    ~~ cen 6714   cardccrd 7149
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-er 6511  df-en 6717  df-card 7150
This theorem is referenced by: (None)
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