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Theorem relsdom 7826
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 7825 . 2 Rel ≼
2 reldif 5150 . . 3 (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3 df-sdom 7822 . . . 4 ≺ = ( ≼ ∖ ≈ )
43releqi 5115 . . 3 (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
52, 4sylibr 222 . 2 (Rel ≼ → Rel ≺ )
61, 5ax-mp 5 1 Rel ≺
Colors of variables: wff setvar class
Syntax hints:  cdif 3536  Rel wrel 5033  cen 7816  cdom 7817  csdm 7818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034  df-rel 5035  df-dom 7821  df-sdom 7822
This theorem is referenced by:  domdifsn  7906  sdom0  7955  sdomirr  7960  sdomdif  7971  sucdom2  8019  sdom1  8023  unxpdom  8030  unxpdom2  8031  sucxpdom  8032  isfinite2  8081  fin2inf  8086  card2on  8320  cdaxpdom  8872  cdafi  8873  cfslb2n  8951  isfin5  8982  isfin6  8983  isfin4-3  8998  fin56  9076  fin67  9078  sdomsdomcard  9239  gchi  9303  canthp1lem1  9331  canthp1lem2  9332  canthp1  9333  frgpnabl  18050  fphpd  36222
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