MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relen Unicode version

Theorem relen 6868
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.
StepHypRef Expression
1 df-en 6864 . 2  |-  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
21relopabi 4811 1  |-  Rel  ~~
Colors of variables: wff set class
Syntax hints:   E.wex 1528   Rel wrel 4694   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  bren  6871  isfi  6885  enssdom  6886  ener  6908  sbthcl  6983  xpen  7024  pwen  7034  php3  7047  f1finf1o  7086  isnum2  7578  inffien  7690  cdaen  7799  cdaenun  7800  cdainflem  7817  cdalepw  7822  infmap2  7844  fin4i  7924  fin4en1  7935  isfin4-3  7941  enfin2i  7947  fin45  8018  axcc3  8064  engch  8250  hargch  8299  hasheni  11347  frgpcyg  16527  ctbnfien  26901  enfixsn  27257  mapfien2  27258  lbslcic  27311  en1uniel  27380  pmtrfv  27395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-en 6864
  Copyright terms: Public domain W3C validator