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Theorem relsdom 7145
Description: Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
relsdom  |-  Rel  ~<

Proof of Theorem relsdom
StepHypRef Expression
1 reldom 7144 . 2  |-  Rel  ~<_
2 reldif 5023 . . 3  |-  ( Rel  ~<_  ->  Rel  (  ~<_  \  ~~  ) )
3 df-sdom 7141 . . . 4  |-  ~<  =  (  ~<_  \  ~~  )
43releqi 4989 . . 3  |-  ( Rel 
~< 
<->  Rel  (  ~<_  \  ~~  ) )
52, 4sylibr 205 . 2  |-  ( Rel  ~<_  ->  Rel  ~<  )
61, 5ax-mp 5 1  |-  Rel  ~<
Colors of variables: wff set class
Syntax hints:    \ cdif 3303   Rel wrel 4912    ~~ cen 7135    ~<_ cdom 7136    ~< csdm 7137
This theorem is referenced by:  domdifsn  7220  sdom0  7268  sdomirr  7273  sdomdif  7284  sucdom2  7332  sdom1  7337  unxpdom  7345  unxpdom2  7346  sucxpdom  7347  isfinite2  7394  fin2inf  7399  card2on  7551  cdaxpdom  8100  cdafi  8101  cfslb2n  8179  isfin5  8210  isfin6  8211  isfin4-3  8226  fin56  8304  fin67  8306  sdomsdomcard  8466  gchi  8530  canthp1lem1  8558  canthp1lem2  8559  canthp1  8560  frgpnabl  15517  fphpd  26915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-opab 4292  df-xp 4913  df-rel 4914  df-dom 7140  df-sdom 7141
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