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Theorem enrefg 7098
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 5672 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 7083 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1268 . 2  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~~  A )
43anidms 627 1  |-  ( A  e.  V  ->  A  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   class class class wbr 4172    _I cid 4453    |` cres 4839   -1-1-onto->wf1o 5412    ~~ cen 7065
This theorem is referenced by:  enref  7099  eqeng  7100  domrefg  7101  difsnen  7149  sdomirr  7203  mapdom1  7231  mapdom2  7237  onfin  7256  ssnnfi  7287  infdifsn  7567  infdiffi  7568  onenon  7792  cardonle  7800  cda1en  8011  xpcdaen  8019  mapcdaen  8020  onacda  8033  ssfin4  8146  canthp1lem1  8483  gchhar  8502  hashfac  11662  mreexexlem3d  13826  cyggenod  15449  fidomndrnglem  16321  frlmpwfi  27130  fiuneneq  27381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7069
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