Theorem relen 6631

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 6628	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
2	1	relopabi 4660	1 ⊢ Rel ≈

Syntax hints:  ∃wex 1468  Rel wrel 4539  –1-1-onto→wf1o 5117   ≈ cen 6625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-rel 4541  df-en 6628

This theorem is referenced by:  encv  6633  isfi  6648  enssdom  6649  ener  6666  en1uniel  6691  xpen  6732  enomnilem  7003  djuenun  7061  pwf1oexmid  13183