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Theorem relsdom 7848
 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 7847 . 2 Rel ≼
2 reldif 5161 . . 3 (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3 df-sdom 7844 . . . 4 ≺ = ( ≼ ∖ ≈ )
43releqi 5125 . . 3 (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
52, 4sylibr 223 . 2 (Rel ≼ → Rel ≺ )
61, 5ax-mp 5 1 Rel ≺
 Colors of variables: wff setvar class Syntax hints:   ∖ cdif 3537  Rel wrel 5043   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-dom 7843  df-sdom 7844 This theorem is referenced by:  domdifsn  7928  sdom0  7977  sdomirr  7982  sdomdif  7993  sucdom2  8041  sdom1  8045  unxpdom  8052  unxpdom2  8053  sucxpdom  8054  isfinite2  8103  fin2inf  8108  card2on  8342  cdaxpdom  8894  cdafi  8895  cfslb2n  8973  isfin5  9004  isfin6  9005  isfin4-3  9020  fin56  9098  fin67  9100  sdomsdomcard  9261  gchi  9325  canthp1lem1  9353  canthp1lem2  9354  canthp1  9355  frgpnabl  18101  fphpd  36398
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