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Theorem relsdom 8004
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 8003 . 2 Rel ≼
2 reldif 5271 . . 3 (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3 df-sdom 8000 . . . 4 ≺ = ( ≼ ∖ ≈ )
43releqi 5236 . . 3 (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
52, 4sylibr 224 . 2 (Rel ≼ → Rel ≺ )
61, 5ax-mp 5 1 Rel ≺
Colors of variables: wff setvar class
Syntax hints:  cdif 3604  Rel wrel 5148  cen 7994  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-dom 7999  df-sdom 8000
This theorem is referenced by:  domdifsn  8084  sdom0  8133  sdomirr  8138  sdomdif  8149  sucdom2  8197  sdom1  8201  unxpdom  8208  unxpdom2  8209  sucxpdom  8210  isfinite2  8259  fin2inf  8264  card2on  8500  cdaxpdom  9049  cdafi  9050  cfslb2n  9128  isfin5  9159  isfin6  9160  isfin4-3  9175  fin56  9253  fin67  9255  sdomsdomcard  9420  gchi  9484  canthp1lem1  9512  canthp1lem2  9513  canthp1  9514  frgpnabl  18324  fphpd  37697
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