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Theorem enrefg 6895
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 5513 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 6880 . . 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 1686   class class class wbr 4025    _I cid 4306    |` cres 4693   -1-1-onto->wf1o 5256    ~~ cen 6862
This theorem is referenced by:  enref  6896  eqeng  6897  domrefg  6898  difsnen  6946  sdomirr  7000  mapdom1  7028  mapdom2  7034  onfin  7053  ssnnfi  7084  infdifsn  7359  infdiffi  7360  onenon  7584  cardonle  7592  cda1en  7803  xpcdaen  7811  mapcdaen  7812  onacda  7825  ssfin4  7938  canthp1lem1  8276  gchhar  8295  hashfac  11398  mreexexlem3d  13550  cyggenod  15173  fidomndrnglem  16049  frlmpwfi  27273  fiuneneq  27524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-en 6866
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