Theorem relen 6771

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 6768	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
2	1	relopabi 4770	1 |- Rel ~~

Syntax hints:   E.wex 1503   Rel wrel 4649   -1-1-onto->wf1o 5234   ~~ cen 6765

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650  df-rel 4651  df-en 6768

This theorem is referenced by:  encv  6773  isfi  6788  enssdom  6789  ener  6806  en1uniel  6831  xpen  6874  enomnilem  7167  enmkvlem  7190  enwomnilem  7198  djuenun  7242  cc3  7298  pwf1oexmid  15228