Theorem relen 7846

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 7842	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
2	1	relopabi 5167	1 ⊢ Rel ≈

Syntax hints:  ∃wex 1695  Rel wrel 5043  –1-1-onto→wf1o 5803   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-en 7842

This theorem is referenced by:  encv  7849  isfi  7865  enssdom  7866  ener  7888  enerOLD  7889  en1uniel  7914  enfixsn  7954  sbthcl  7967  xpen  8008  pwen  8018  php3  8031  f1finf1o  8072  mapfien2  8197  isnum2  8654  inffien  8769  cdaen  8878  cdaenun  8879  cdainflem  8896  cdalepw  8901  infmap2  8923  fin4i  9003  fin4en1  9014  isfin4-3  9020  enfin2i  9026  fin45  9097  axcc3  9143  engch  9329  hargch  9374  hasheni  12998  pmtrfv  17695  frgpcyg  19741  lbslcic  19999  phpreu  32563  ctbnfien  36400