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Theorem relsdom 8167
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 8166 . 2 Rel ≼
2 reldif 5407 . . 3 (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3 df-sdom 8163 . . . 4 ≺ = ( ≼ ∖ ≈ )
43releqi 5372 . . 3 (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
52, 4sylibr 225 . 2 (Rel ≼ → Rel ≺ )
61, 5ax-mp 5 1 Rel ≺
Colors of variables: wff setvar class
Syntax hints:  cdif 3729  Rel wrel 5282  cen 8157  cdom 8158  csdm 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283  df-rel 5284  df-dom 8162  df-sdom 8163
This theorem is referenced by:  domdifsn  8250  sdom0  8299  sdomirr  8304  sdomdif  8315  sucdom2  8363  sdom1  8367  unxpdom  8374  unxpdom2  8375  sucxpdom  8376  isfinite2  8425  fin2inf  8430  card2on  8666  cdaxpdom  9264  cdafi  9265  cfslb2n  9343  isfin5  9374  isfin6  9375  isfin4-3  9390  fin56  9468  fin67  9470  sdomsdomcard  9635  gchi  9699  canthp1lem1  9727  canthp1lem2  9728  canthp1  9729  frgpnabl  18544  fphpd  38058
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