Theorem relen 6744

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 6741	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
2	1	relopabi 4753	1 |- Rel ~~

Syntax hints:   E.wex 1492   Rel wrel 4632   -1-1-onto->wf1o 5216   ~~ cen 6738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633  df-rel 4634  df-en 6741

This theorem is referenced by:  encv  6746  isfi  6761  enssdom  6762  ener  6779  en1uniel  6804  xpen  6845  enomnilem  7136  enmkvlem  7159  enwomnilem  7167  djuenun  7211  cc3  7267  pwf1oexmid  14752