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Theorem cardval3 8285
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 3067 . 2  |-  ( A  e.  dom  card  ->  A  e.  _V )
2 isnum2 8278 . . . 4  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
3 rabn0 3758 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  E. x  e.  On  x  ~~  A )
4 intex 4549 . . . 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 4398 . . . . 5  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rabbidv 3050 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  x  ~~  y }  =  {
x  e.  On  |  x  ~~  A } )
98inteqd 4231 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  x  ~~  y }  =  |^| { x  e.  On  |  x  ~~  A } )
10 df-card 8272 . . 3  |-  card  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  x  ~~  y } )
119, 10fvmptg 5886 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
121, 6, 11syl2anc 659 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   {crab 2757   _Vcvv 3058   (/)c0 3737   |^|cint 4226   class class class wbr 4394   Oncon0 4821   dom cdm 4942   ` cfv 5525    ~~ cen 7471   cardccrd 8268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-en 7475  df-card 8272
This theorem is referenced by:  cardid2  8286  oncardval  8288  cardidm  8292  cardne  8298  cardval  8873
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