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Theorem isnum2 7578
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 7576 . . . 4  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
21fdmi 5394 . . 3  |-  dom  card  =  { y  |  E. x  e.  On  x  ~~  y }
32eleq2i 2347 . 2  |-  ( A  e.  dom  card  <->  A  e.  { y  |  E. x  e.  On  x  ~~  y } )
4 relen 6868 . . . . 5  |-  Rel  ~~
54brrelex2i 4730 . . . 4  |-  ( x 
~~  A  ->  A  e.  _V )
65rexlimivw 2663 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
7 breq2 4027 . . . 4  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rexbidv 2564 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  x  ~~  y  <->  E. x  e.  On  x  ~~  A
) )
96, 8elab3 2921 . 2  |-  ( A  e.  { y  |  E. x  e.  On  x  ~~  y }  <->  E. x  e.  On  x  ~~  A
)
103, 9bitri 240 1  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   Oncon0 4392   dom cdm 4689    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  isnumi  7579  ennum  7580  xpnum  7584  cardval3  7585  dfac10c  7764  isfin7-2  8022  numth2  8098  inawinalem  8311
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-fun 5257  df-fn 5258  df-f 5259  df-en 6864  df-card 7572
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