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Theorem enrefg 7362
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 5697 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 7347 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1303 . 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 1756   class class class wbr 4313    _I cid 4652    |` cres 4863   -1-1-onto->wf1o 5438    ~~ cen 7328
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-en 7332
This theorem is referenced by:  enref  7363  eqeng  7364  domrefg  7365  difsnen  7414  sdomirr  7469  mapdom1  7497  mapdom2  7503  onfin  7522  ssnnfi  7553  infdifsn  7883  infdiffi  7884  onenon  8140  cardonle  8148  cda1en  8365  xpcdaen  8373  mapcdaen  8374  onacda  8387  ssfin4  8500  canthp1lem1  8840  gchhar  8867  hashfac  12232  mreexexlem3d  14605  cyggenod  16382  fidomndrnglem  17400  mdetunilem8  18447  frlmpwfi  29479  fiuneneq  29588
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