Theorem relen 9016

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 9012	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
2	1	relopabi 5809	1 ⊢ Rel ≈

This proof depends on syntax axioms:  ∃wex 1812  Rel wrel 5663  ⟶1-1ontowf1o 6565   ≈ cen 9008

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174  df-xp 5664  df-rel 5665  df-en 9012

This theorem is used by:  encv  9019  isfi  9040  enssdomOLD  9042  ener  9066  enfixsn  9143  sbthcl  9156  xpen  9197  pwen  9207  mapfien2  9438  isnum2  10063  inffien  10181  djuen  10287  djuenun  10288  cdainflem  10305  djulepw  10310  infmap2  10334  fin4i  10415  fin4en1  10426  isfin4p1  10432  enfin2i  10438  fin45  10509  axcc3  10555  engch  10752  hargch  10797  hasheni  14533  pmtrfv  19717  frgpcyg  21934  lbslcic  22202  kardenir  35967  phpreu  38795  ctbnfien  44091