Theorem relen 8114

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 8110	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
2	1	relopabi 5389	1 ⊢ Rel ≈

Syntax hints:  ∃wex 1841  Rel wrel 5259  –1-1-onto→wf1o 6036   ≈ cen 8106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-opab 4853  df-xp 5260  df-rel 5261  df-en 8110

This theorem is referenced by:  encv  8117  isfi  8133  enssdom  8134  ener  8156  en1uniel  8181  enfixsn  8222  sbthcl  8235  xpen  8276  pwen  8286  php3  8299  f1finf1o  8340  mapfien2  8467  isnum2  8932  inffien  9047  cdaen  9158  cdaenun  9159  cdainflem  9176  cdalepw  9181  infmap2  9203  fin4i  9283  fin4en1  9294  isfin4-3  9300  enfin2i  9306  fin45  9377  axcc3  9423  engch  9613  hargch  9658  hasheni  13301  pmtrfv  18043  frgpcyg  20095  lbslcic  20353  phpreu  33675  ctbnfien  37853