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Theorem ctex 26008
Description: A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
ctex  |-  ( A  ~<_  om  ->  A  e.  _V )

Proof of Theorem ctex
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brdomi 7321 . 2  |-  ( A  ~<_  om  ->  E. f 
f : A -1-1-> om )
2 f1dm 5610 . . . 4  |-  ( f : A -1-1-> om  ->  dom  f  =  A )
3 vex 2975 . . . . 5  |-  f  e. 
_V
43dmex 6511 . . . 4  |-  dom  f  e.  _V
52, 4syl6eqelr 2532 . . 3  |-  ( f : A -1-1-> om  ->  A  e.  _V )
65exlimiv 1688 . 2  |-  ( E. f  f : A -1-1-> om 
->  A  e.  _V )
71, 6syl 16 1  |-  ( A  ~<_  om  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1586    e. wcel 1756   _Vcvv 2972   class class class wbr 4292   dom cdm 4840   -1-1->wf1 5415   omcom 6476    ~<_ cdom 7308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-dm 4850  df-rn 4851  df-fn 5421  df-f 5422  df-f1 5423  df-dom 7312
This theorem is referenced by:  ssct  26009  xpct  26010  fnct  26013  dmct  26014  cnvct  26015  fimact  26017  mptct  26018  mptctf  26021  gsummpt2co  26249  elsigagen2  26591  measvunilem  26626  measvunilem0  26627  measvuni  26628  sxbrsigalem1  26700
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