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Theorem isnum2 8343
Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
isnum2  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
Distinct variable group:    x, A

Proof of Theorem isnum2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardf2 8341 . . . 4  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
21fdmi 5742 . . 3  |-  dom  card  =  { y  |  E. x  e.  On  x  ~~  y }
32eleq2i 2535 . 2  |-  ( A  e.  dom  card  <->  A  e.  { y  |  E. x  e.  On  x  ~~  y } )
4 relen 7540 . . . . 5  |-  Rel  ~~
54brrelex2i 5050 . . . 4  |-  ( x 
~~  A  ->  A  e.  _V )
65rexlimivw 2946 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
7 breq2 4460 . . . 4  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rexbidv 2968 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  x  ~~  y  <->  E. x  e.  On  x  ~~  A
) )
96, 8elab3 3253 . 2  |-  ( A  e.  { y  |  E. x  e.  On  x  ~~  y }  <->  E. x  e.  On  x  ~~  A
)
103, 9bitri 249 1  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   class class class wbr 4456   Oncon0 4887   dom cdm 5008    ~~ cen 7532   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596  df-fn 5597  df-f 5598  df-en 7536  df-card 8337
This theorem is referenced by:  isnumi  8344  ennum  8345  xpnum  8349  cardval3  8350  dfac10c  8535  isfin7-2  8793  numth2  8868  inawinalem  9084
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