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Theorem enrefg 7068
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 5646 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 7053 . . 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 1717   class class class wbr 4146    _I cid 4427    |` cres 4813   -1-1-onto->wf1o 5386    ~~ cen 7035
This theorem is referenced by:  enref  7069  eqeng  7070  domrefg  7071  difsnen  7119  sdomirr  7173  mapdom1  7201  mapdom2  7207  onfin  7226  ssnnfi  7257  infdifsn  7537  infdiffi  7538  onenon  7762  cardonle  7770  cda1en  7981  xpcdaen  7989  mapcdaen  7990  onacda  8003  ssfin4  8116  canthp1lem1  8453  gchhar  8472  hashfac  11627  mreexexlem3d  13791  cyggenod  15414  fidomndrnglem  16286  frlmpwfi  26924  fiuneneq  27175
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-en 7039
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