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Theorem enrefg 7626
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 5872 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 f1oen2g 7611 . . 3  |-  ( ( A  e.  V  /\  A  e.  V  /\  (  _I  |`  A ) : A -1-1-onto-> A )  ->  A  ~~  A )
31, 2mp3an3 1362 . 2  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~~  A )
43anidms 655 1  |-  ( A  e.  V  ->  A  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1897   class class class wbr 4415    _I cid 4762    |` cres 4854   -1-1-onto->wf1o 5599    ~~ cen 7591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-en 7595
This theorem is referenced by:  enref  7627  eqeng  7628  domrefg  7629  difsnen  7679  sdomirr  7734  mapdom1  7762  mapdom2  7768  onfin  7788  ssnnfi  7816  infdifsn  8187  infdiffi  8188  onenon  8408  cardonle  8416  cda1en  8630  xpcdaen  8638  mapcdaen  8639  onacda  8652  ssfin4  8765  canthp1lem1  9102  gchhar  9129  hashfac  12653  mreexexlem3d  15600  cyggenod  17567  fidomndrnglem  18578  mdetunilem8  19692  frlmpwfi  36000  fiuneneq  36115
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