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Theorem relsdom 7420
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 7419 . 2  |-  Rel  ~<_
2 reldif 5060 . . 3  |-  ( Rel  ~<_  ->  Rel  (  ~<_  \  ~~  ) )
3 df-sdom 7416 . . . 4  |-  ~<  =  (  ~<_  \  ~~  )
43releqi 5024 . . 3  |-  ( Rel 
~< 
<->  Rel  (  ~<_  \  ~~  ) )
52, 4sylibr 212 . 2  |-  ( Rel  ~<_  ->  Rel  ~<  )
61, 5ax-mp 5 1  |-  Rel  ~<
Colors of variables: wff setvar class
Syntax hints:    \ cdif 3426   Rel wrel 4946    ~~ cen 7410    ~<_ cdom 7411    ~< csdm 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-opab 4452  df-xp 4947  df-rel 4948  df-dom 7415  df-sdom 7416
This theorem is referenced by:  domdifsn  7497  sdom0  7546  sdomirr  7551  sdomdif  7562  sucdom2  7611  sdom1  7616  unxpdom  7624  unxpdom2  7625  sucxpdom  7626  isfinite2  7674  fin2inf  7679  card2on  7873  cdaxpdom  8462  cdafi  8463  cfslb2n  8541  isfin5  8572  isfin6  8573  isfin4-3  8588  fin56  8666  fin67  8668  sdomsdomcard  8828  gchi  8895  canthp1lem1  8923  canthp1lem2  8924  canthp1  8925  frgpnabl  16466  fphpd  29296
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