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Theorem cardval3 7773
Description: An alternative definition of the value of  ( card `  A
) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
cardval3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Distinct variable group:    x, A

Proof of Theorem cardval3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . 2  |-  ( A  e.  dom  card  ->  A  e.  _V )
2 isnum2 7766 . . . 4  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
3 rabn0 3591 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  E. x  e.  On  x  ~~  A )
4 intex 4298 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
52, 3, 43bitr2i 265 . . 3  |-  ( A  e.  dom  card  <->  |^| { x  e.  On  |  x  ~~  A }  e.  _V )
65biimpi 187 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
7 breq2 4158 . . . . 5  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rabbidv 2892 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  x  ~~  y }  =  {
x  e.  On  |  x  ~~  A } )
98inteqd 3998 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  x  ~~  y }  =  |^| { x  e.  On  |  x  ~~  A } )
10 df-card 7760 . . 3  |-  card  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  x  ~~  y } )
119, 10fvmptg 5744 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
121, 6, 11syl2anc 643 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   {crab 2654   _Vcvv 2900   (/)c0 3572   |^|cint 3993   class class class wbr 4154   Oncon0 4523   dom cdm 4819   ` cfv 5395    ~~ cen 7043   cardccrd 7756
This theorem is referenced by:  cardid2  7774  oncardval  7776  cardidm  7780  cardne  7786  cardval  8355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-en 7047  df-card 7760
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