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Theorem enrefg 7559
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
enrefg  |-  ( A  e.  V  ->  A  ~~  A )

Proof of Theorem enrefg
StepHypRef Expression
1 f1oi 5857 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 7544 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1313 . 2  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~~  A )
43anidms 645 1  |-  ( A  e.  V  ->  A  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   class class class wbr 4453    _I cid 4796    |` cres 5007   -1-1-onto->wf1o 5593    ~~ cen 7525
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-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  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-f1 5599  df-fo 5600  df-f1o 5601  df-en 7529
This theorem is referenced by:  enref  7560  eqeng  7561  domrefg  7562  difsnen  7611  sdomirr  7666  mapdom1  7694  mapdom2  7700  onfin  7720  ssnnfi  7751  infdifsn  8085  infdiffi  8086  onenon  8342  cardonle  8350  cda1en  8567  xpcdaen  8575  mapcdaen  8576  onacda  8589  ssfin4  8702  canthp1lem1  9042  gchhar  9069  hashfac  12488  mreexexlem3d  14918  cyggenod  16760  fidomndrnglem  17825  mdetunilem8  18990  frlmpwfi  30974  fiuneneq  31083
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