Theorem relen 6904

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 6901	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
2	1	relopabi 4850	1 |- Rel ~~

Syntax hints:   E.wex 1538   Rel wrel 4725   -1-1-onto->wf1o 5320   ~~ cen 6898

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4726  df-rel 4727  df-en 6901

This theorem is referenced by:  encv  6906  isfi  6925  enssdom  6926  ener  6944  en1uniel  6969  xpen  7019  enomnilem  7321  enmkvlem  7344  enwomnilem  7352  djuenun  7410  cc3  7470  pwf1oexmid  16478