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Theorem relsdom 7013
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 7012 . 2  |-  Rel  ~<_
2 reldif 4908 . . 3  |-  ( Rel  ~<_  ->  Rel  (  ~<_  \  ~~  ) )
3 df-sdom 7009 . . . 4  |-  ~<  =  (  ~<_  \  ~~  )
43releqi 4875 . . 3  |-  ( Rel 
~< 
<->  Rel  (  ~<_  \  ~~  ) )
52, 4sylibr 203 . 2  |-  ( Rel  ~<_  ->  Rel  ~<  )
61, 5ax-mp 8 1  |-  Rel  ~<
Colors of variables: wff set class
Syntax hints:    \ cdif 3235   Rel wrel 4797    ~~ cen 7003    ~<_ cdom 7004    ~< csdm 7005
This theorem is referenced by:  domdifsn  7088  sdom0  7136  sdomirr  7141  sdomdif  7152  sucdom2  7200  sdom1  7205  unxpdom  7213  unxpdom2  7214  sucxpdom  7215  isfinite2  7262  fin2inf  7267  card2on  7415  cdaxpdom  7962  cdafi  7963  cfslb2n  8041  isfin5  8072  isfin6  8073  isfin4-3  8088  fin56  8166  fin67  8168  sdomsdomcard  8329  gchi  8393  canthp1lem1  8421  canthp1lem2  8422  canthp1  8423  frgpnabl  15373  fphpd  26405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-opab 4180  df-xp 4798  df-rel 4799  df-dom 7008  df-sdom 7009
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