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Theorem relsdom 8510
 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 8509 . 2 Rel ≼
2 reldif 5687 . . 3 (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3 df-sdom 8506 . . . 4 ≺ = ( ≼ ∖ ≈ )
43releqi 5651 . . 3 (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
52, 4sylibr 235 . 2 (Rel ≼ → Rel ≺ )
61, 5ax-mp 5 1 Rel ≺
 Colors of variables: wff setvar class Syntax hints:   ∖ cdif 3937  Rel wrel 5559   ≈ cen 8500   ≼ cdom 8501   ≺ csdm 8502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-opab 5126  df-xp 5560  df-rel 5561  df-dom 8505  df-sdom 8506 This theorem is referenced by:  domdifsn  8594  sdom0  8643  sdomirr  8648  sdomdif  8659  sucdom2  8708  sdom1  8712  unxpdom  8719  unxpdom2  8720  sucxpdom  8721  isfinite2  8770  fin2inf  8775  card2on  9012  djuxpdom  9605  djufi  9606  infdif  9625  cfslb2n  9684  isfin5  9715  isfin6  9716  isfin4p1  9731  fin56  9809  fin67  9811  sdomsdomcard  9976  gchi  10040  canthp1lem1  10068  canthp1lem2  10069  canthp1  10070  frgpnabl  18931  fphpd  39297
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