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Theorem enrefg 8530
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 6649 . . 3 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 f1oen2g 8515 . . 3 ((𝐴𝑉𝐴𝑉 ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝐴𝐴)
31, 2mp3an3 1443 . 2 ((𝐴𝑉𝐴𝑉) → 𝐴𝐴)
43anidms 567 1 (𝐴𝑉𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5063   I cid 5458  cres 5556  1-1-ontowf1o 6351  cen 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-en 8499
This theorem is referenced by:  enref  8531  eqeng  8532  domrefg  8533  difsnen  8588  sdomirr  8643  mapdom1  8671  mapdom2  8677  onfin  8698  ssnnfi  8726  rneqdmfinf1o  8789  infdifsn  9109  infdiffi  9110  onenon  9367  cardonle  9375  dju1en  9586  xpdjuen  9594  mapdjuen  9595  onadju  9608  ssfin4  9721  canthp1lem1  10063  gchhar  10090  hashfac  13806  mreexexlem3d  16907  cyggenod  18923  fidomndrnglem  19998  mdetunilem8  21144  frlmpwfi  39563  fiuneneq  39662  enrelmap  40208
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