Theorem relen 6854

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 6851	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
2	1	relopabi 4821	1 |- Rel ~~

Syntax hints:   E.wex 1516   Rel wrel 4698   -1-1-onto->wf1o 5289   ~~ cen 6848

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-xp 4699  df-rel 4700  df-en 6851

This theorem is referenced by:  encv  6856  isfi  6875  enssdom  6876  ener  6894  en1uniel  6919  xpen  6967  enomnilem  7266  enmkvlem  7289  enwomnilem  7297  djuenun  7355  cc3  7415  pwf1oexmid  16138