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Theorem isnum2 8338
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 8336 . . . 4  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
21fdmi 5742 . . 3  |-  dom  card  =  { y  |  E. x  e.  On  x  ~~  y }
32eleq2i 2545 . 2  |-  ( A  e.  dom  card  <->  A  e.  { y  |  E. x  e.  On  x  ~~  y } )
4 relen 7533 . . . . 5  |-  Rel  ~~
54brrelex2i 5047 . . . 4  |-  ( x 
~~  A  ->  A  e.  _V )
65rexlimivw 2956 . . 3  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
7 breq2 4457 . . . 4  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rexbidv 2978 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  x  ~~  y  <->  E. x  e.  On  x  ~~  A
) )
96, 8elab3 3262 . 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 1379    e. wcel 1767   {cab 2452   E.wrex 2818   _Vcvv 3118   class class class wbr 4453   Oncon0 4884   dom cdm 5005    ~~ cen 7525   cardccrd 8328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596  df-fn 5597  df-f 5598  df-en 7529  df-card 8332
This theorem is referenced by:  isnumi  8339  ennum  8340  xpnum  8344  cardval3  8345  dfac10c  8530  isfin7-2  8788  numth2  8863  inawinalem  9079
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