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Theorem dfsdom2 8628
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 8500 . 2 ≺ = ( ≼ ∖ ≈ )
2 sbthcl 8627 . . 3 ≈ = ( ≼ ∩ ≼ )
32difeq2i 4093 . 2 ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ≼ ))
4 difin 4235 . 2 ( ≼ ∖ ( ≼ ∩ ≼ )) = ( ≼ ∖ ≼ )
51, 3, 43eqtri 2845 1 ≺ = ( ≼ ∖ ≼ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cdif 3930  cin 3932  ccnv 5547  cen 8494  cdom 8495  csdm 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500
This theorem is referenced by:  brsdom2  8629
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