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Theorem enrefg 6893
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 5511 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 6878 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1266 . 2  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~~  A )
43anidms 626 1  |-  ( A  e.  V  ->  A  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   class class class wbr 4023    _I cid 4304    |` cres 4691   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  enref  6894  eqeng  6895  domrefg  6896  difsnen  6944  sdomirr  6998  mapdom1  7026  mapdom2  7032  onfin  7051  ssnnfi  7082  infdifsn  7357  infdiffi  7358  onenon  7582  cardonle  7590  cda1en  7801  xpcdaen  7809  mapcdaen  7810  onacda  7823  ssfin4  7936  canthp1lem1  8274  gchhar  8293  hashfac  11396  mreexexlem3d  13548  cyggenod  15171  fidomndrnglem  16047  frlmpwfi  27262  fiuneneq  27513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
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