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Theorem carden2bex 6517
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 6382 . . . . 5  |-  ( A 
~~  B  ->  (
y  ~~  A  <->  y  ~~  B ) )
21rabbidv 2594 . . . 4  |-  ( A 
~~  B  ->  { y  e.  On  |  y 
~~  A }  =  { y  e.  On  |  y  ~~  B }
)
32inteqd 3649 . . 3  |-  ( A 
~~  B  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
43adantr 270 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  |^| { y  e.  On  |  y 
~~  A }  =  |^| { y  e.  On  |  y  ~~  B }
)
5 cardval3ex 6513 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
65adantl 271 . 2  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
7 entr 6331 . . . . . 6  |-  ( ( x  ~~  A  /\  A  ~~  B )  ->  x  ~~  B )
87expcom 114 . . . . 5  |-  ( A 
~~  B  ->  (
x  ~~  A  ->  x 
~~  B ) )
98reximdv 2463 . . . 4  |-  ( A 
~~  B  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  On  x  ~~  B
) )
109imp 122 . . 3  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  E. x  e.  On  x  ~~  B
)
11 cardval3ex 6513 . . 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 2124 1  |-  ( ( A  ~~  B  /\  E. x  e.  On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wrex 2350   {crab 2353   |^|cint 3644   class class class wbr 3793   Oncon0 4126   ` cfv 4932    ~~ 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-13 1445  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  ax-un 4196
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-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-er 6172  df-en 6288  df-card 6508
This theorem is referenced by: (None)
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