Theorem relen 7960

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 7956	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
2	1	relopabi 5245	1 |- Rel ~~

Syntax hints:   E.wex 1704   Rel wrel 5119   -1-1-onto->wf1o 5887   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-en 7956

This theorem is referenced by:  encv  7963  isfi  7979  enssdom  7980  ener  8002  enerOLD  8003  en1uniel  8028  enfixsn  8069  sbthcl  8082  xpen  8123  pwen  8133  php3  8146  f1finf1o  8187  mapfien2  8314  isnum2  8771  inffien  8886  cdaen  8995  cdaenun  8996  cdainflem  9013  cdalepw  9018  infmap2  9040  fin4i  9120  fin4en1  9131  isfin4-3  9137  enfin2i  9143  fin45  9214  axcc3  9260  engch  9450  hargch  9495  hasheni  13136  pmtrfv  17872  frgpcyg  19922  lbslcic  20180  phpreu  33393  ctbnfien  37382