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Theorem dfsdom2 7946
Description: Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
dfsdom2 ≺ = ( ≼ ∖ ≼ )

Proof of Theorem dfsdom2
StepHypRef Expression
1 df-sdom 7822 . 2 ≺ = ( ≼ ∖ ≈ )
2 sbthcl 7945 . . 3 ≈ = ( ≼ ∩ ≼ )
32difeq2i 3687 . 2 ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ≼ ))
4 difin 3823 . 2 ( ≼ ∖ ( ≼ ∩ ≼ )) = ( ≼ ∖ ≼ )
51, 3, 43eqtri 2636 1 ≺ = ( ≼ ∖ ≼ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cdif 3537  cin 3539  ccnv 5027  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 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  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-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822
This theorem is referenced by:  brsdom2  7947
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