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Theorem enrefg 7932
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 (𝐴𝑉𝐴𝐴)

Proof of Theorem enrefg
StepHypRef Expression
1 f1oi 6133 . . 3 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 f1oen2g 7917 . . 3 ((𝐴𝑉𝐴𝑉 ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝐴𝐴)
31, 2mp3an3 1410 . 2 ((𝐴𝑉𝐴𝑉) → 𝐴𝐴)
43anidms 676 1 (𝐴𝑉𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992   class class class wbr 4618   I cid 4989  cres 5081  1-1-ontowf1o 5849  cen 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-en 7901
This theorem is referenced by:  enref  7933  eqeng  7934  domrefg  7935  difsnen  7987  sdomirr  8042  mapdom1  8070  mapdom2  8076  onfin  8096  ssnnfi  8124  rneqdmfinf1o  8187  infdifsn  8499  infdiffi  8500  onenon  8720  cardonle  8728  cda1en  8942  xpcdaen  8950  mapcdaen  8951  onacda  8964  ssfin4  9077  canthp1lem1  9419  gchhar  9446  hashfac  13177  mreexexlem3d  16222  cyggenod  18202  fidomndrnglem  19220  mdetunilem8  20339  frlmpwfi  37134  fiuneneq  37242  enrelmap  37759
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