Theorem relen 8514

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.

Step	Hyp	Ref	Expression
1		df-en 8510	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
2	1	relopabi 5694	1 ⊢ Rel ≈

Syntax hints:  ∃wex 1780  Rel wrel 5560  –1-1-onto→wf1o 6354   ≈ cen 8506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561  df-rel 5562  df-en 8510

This theorem is referenced by:  encv  8517  isfi  8533  enssdom  8534  ener  8556  en1uniel  8581  enfixsn  8626  sbthcl  8639  xpen  8680  pwen  8690  php3  8703  f1finf1o  8745  mapfien2  8872  isnum2  9374  inffien  9489  djuen  9595  djuenun  9596  cdainflem  9613  djulepw  9618  infmap2  9640  fin4i  9720  fin4en1  9731  isfin4p1  9737  enfin2i  9743  fin45  9814  axcc3  9860  engch  10050  hargch  10095  hasheni  13709  pmtrfv  18580  frgpcyg  20720  lbslcic  20985  phpreu  34891  ctbnfien  39435