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Theorem isnum2 7594
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 7592 . . . 4  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
21fdmi 5410 . . 3  |-  dom  card  =  { y  |  E. x  e.  On  x  ~~  y }
32eleq2i 2360 . 2  |-  ( A  e.  dom  card  <->  A  e.  { y  |  E. x  e.  On  x  ~~  y } )
4 relen 6884 . . . . 5  |-  Rel  ~~
54brrelex2i 4746 . . . 4  |-  ( x 
~~  A  ->  A  e.  _V )
65rexlimivw 2676 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
7 breq2 4043 . . . 4  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rexbidv 2577 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  x  ~~  y  <->  E. x  e.  On  x  ~~  A
) )
96, 8elab3 2934 . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   class class class wbr 4039   Oncon0 4408   dom cdm 4705    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  isnumi  7595  ennum  7596  xpnum  7600  cardval3  7601  dfac10c  7780  isfin7-2  8038  numth2  8114  inawinalem  8327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-en 6880  df-card 7588
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