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Theorem isnum2 7832
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 7830 . . . 4  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
21fdmi 5596 . . 3  |-  dom  card  =  { y  |  E. x  e.  On  x  ~~  y }
32eleq2i 2500 . 2  |-  ( A  e.  dom  card  <->  A  e.  { y  |  E. x  e.  On  x  ~~  y } )
4 relen 7114 . . . . 5  |-  Rel  ~~
54brrelex2i 4919 . . . 4  |-  ( x 
~~  A  ->  A  e.  _V )
65rexlimivw 2826 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
7 breq2 4216 . . . 4  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rexbidv 2726 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  x  ~~  y  <->  E. x  e.  On  x  ~~  A
) )
96, 8elab3 3089 . 2  |-  ( A  e.  { y  |  E. x  e.  On  x  ~~  y }  <->  E. x  e.  On  x  ~~  A
)
103, 9bitri 241 1  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956   class class class wbr 4212   Oncon0 4581   dom cdm 4878    ~~ cen 7106   cardccrd 7822
This theorem is referenced by:  isnumi  7833  ennum  7834  xpnum  7838  cardval3  7839  dfac10c  8018  isfin7-2  8276  numth2  8351  inawinalem  8564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-en 7110  df-card 7826
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