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Theorem relen 8503
 Description: Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
relen Rel ≈

Proof of Theorem relen
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-en 8499 . 2 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
21relopabi 5693 1 Rel ≈
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1773  Rel wrel 5559  –1-1-onto→wf1o 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 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-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  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-sn 4565  df-pr 4567  df-op 4571  df-opab 5126  df-xp 5560  df-rel 5561  df-en 8499 This theorem is referenced by:  encv  8506  isfi  8522  enssdom  8523  ener  8545  en1uniel  8570  enfixsn  8615  sbthcl  8628  xpen  8669  pwen  8679  php3  8692  f1finf1o  8734  mapfien2  8861  isnum2  9363  inffien  9478  djuen  9584  djuenun  9585  cdainflem  9602  djulepw  9607  infmap2  9629  fin4i  9709  fin4en1  9720  isfin4p1  9726  enfin2i  9732  fin45  9803  axcc3  9849  engch  10039  hargch  10084  hasheni  13698  pmtrfv  18500  frgpcyg  20636  lbslcic  20901  phpreu  34743  ctbnfien  39280
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