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Theorem enrefg 6779
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 5368 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 6764 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1271 . 2  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~~  A )
43anidms 629 1  |-  ( A  e.  V  ->  A  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   class class class wbr 3920    _I cid 4197    |` cres 4582   -1-1-onto->wf1o 4591    ~~ cen 6746
This theorem is referenced by:  enref  6780  eqeng  6781  domrefg  6782  difsnen  6829  sdomirr  6883  mapdom1  6911  mapdom2  6917  onfin  6936  ssnnfi  6967  infdifsn  7241  infdiffi  7242  onenon  7466  cardonle  7474  cda1en  7685  xpcdaen  7693  mapcdaen  7694  onacda  7707  ssfin4  7820  canthp1lem1  8154  gchhar  8173  hashfac  11273  cyggenod  15006  fidomndrnglem  15879  frlmpwfi  26428  fiuneneq  26679
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-en 6750
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