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Theorem relen 7561
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 7557 . 2  |-  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
21relopabi 4950 1  |-  Rel  ~~
Colors of variables: wff setvar class
Syntax hints:   E.wex 1635   Rel wrel 4830   -1-1-onto->wf1o 5570    ~~ cen 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-opab 4456  df-xp 4831  df-rel 4832  df-en 7557
This theorem is referenced by:  encv  7564  isfi  7579  enssdom  7580  ener  7602  en1uniel  7627  enfixsn  7666  sbthcl  7679  xpen  7720  pwen  7730  php3  7743  f1finf1o  7783  mapfien2  7904  isnum2  8360  inffien  8478  cdaen  8587  cdaenun  8588  cdainflem  8605  cdalepw  8610  infmap2  8632  fin4i  8712  fin4en1  8723  isfin4-3  8729  enfin2i  8735  fin45  8806  axcc3  8852  engch  9038  hargch  9083  hasheni  12470  pmtrfv  16803  frgpcyg  18912  lbslcic  19170  ctbnfien  35126  mapfien2OLD  35417
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