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Theorem cardval3 7585
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 2796 . 2  |-  ( A  e.  dom  card  ->  A  e.  _V )
2 isnum2 7578 . . . 4  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
3 rabn0 3474 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  E. x  e.  On  x  ~~  A )
4 intex 4167 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
52, 3, 43bitr2i 264 . . 3  |-  ( A  e.  dom  card  <->  |^| { x  e.  On  |  x  ~~  A }  e.  _V )
65biimpi 186 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
7 breq2 4027 . . . . 5  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rabbidv 2780 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  x  ~~  y }  =  {
x  e.  On  |  x  ~~  A } )
98inteqd 3867 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  x  ~~  y }  =  |^| { x  e.  On  |  x  ~~  A } )
10 df-card 7572 . . 3  |-  card  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  x  ~~  y } )
119, 10fvmptg 5600 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
121, 6, 11syl2anc 642 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788   (/)c0 3455   |^|cint 3862   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  cardid2  7586  oncardval  7588  cardidm  7592  cardne  7598  cardval  8168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-en 6864  df-card 7572
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