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Theorem relsdom 7075
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 7074 . 2  |-  Rel  ~<_
2 reldif 4953 . . 3  |-  ( Rel  ~<_  ->  Rel  (  ~<_  \  ~~  ) )
3 df-sdom 7071 . . . 4  |-  ~<  =  (  ~<_  \  ~~  )
43releqi 4919 . . 3  |-  ( Rel 
~< 
<->  Rel  (  ~<_  \  ~~  ) )
52, 4sylibr 204 . 2  |-  ( Rel  ~<_  ->  Rel  ~<  )
61, 5ax-mp 8 1  |-  Rel  ~<
Colors of variables: wff set class
Syntax hints:    \ cdif 3277   Rel wrel 4842    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067
This theorem is referenced by:  domdifsn  7150  sdom0  7198  sdomirr  7203  sdomdif  7214  sucdom2  7262  sdom1  7267  unxpdom  7275  unxpdom2  7276  sucxpdom  7277  isfinite2  7324  fin2inf  7329  card2on  7478  cdaxpdom  8025  cdafi  8026  cfslb2n  8104  isfin5  8135  isfin6  8136  isfin4-3  8151  fin56  8229  fin67  8231  sdomsdomcard  8391  gchi  8455  canthp1lem1  8483  canthp1lem2  8484  canthp1  8485  frgpnabl  15441  fphpd  26767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-rel 4844  df-dom 7070  df-sdom 7071
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