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Theorem cardval3 8237
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 3087 . 2  |-  ( A  e.  dom  card  ->  A  e.  _V )
2 isnum2 8230 . . . 4  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
3 rabn0 3768 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  E. x  e.  On  x  ~~  A )
4 intex 4559 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
52, 3, 43bitr2i 273 . . 3  |-  ( A  e.  dom  card  <->  |^| { x  e.  On  |  x  ~~  A }  e.  _V )
65biimpi 194 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
7 breq2 4407 . . . . 5  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rabbidv 3070 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  x  ~~  y }  =  {
x  e.  On  |  x  ~~  A } )
98inteqd 4244 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  x  ~~  y }  =  |^| { x  e.  On  |  x  ~~  A } )
10 df-card 8224 . . 3  |-  card  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  x  ~~  y } )
119, 10fvmptg 5884 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
121, 6, 11syl2anc 661 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   {crab 2803   _Vcvv 3078   (/)c0 3748   |^|cint 4239   class class class wbr 4403   Oncon0 4830   dom cdm 4951   ` cfv 5529    ~~ cen 7420   cardccrd 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-en 7424  df-card 8224
This theorem is referenced by:  cardid2  8238  oncardval  8240  cardidm  8244  cardne  8250  cardval  8825
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