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Theorem sdomnsym 9049
Description: Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
sdomnsym (𝐴𝐵 → ¬ 𝐵𝐴)

Proof of Theorem sdomnsym
StepHypRef Expression
1 sdomnen 8928 . 2 (𝐴𝐵 → ¬ 𝐴𝐵)
2 sdomdom 8927 . . 3 (𝐴𝐵𝐴𝐵)
3 sdomdom 8927 . . 3 (𝐵𝐴𝐵𝐴)
4 sbth 9044 . . 3 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
52, 3, 4syl2an 597 . 2 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
61, 5mtand 815 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   class class class wbr 5110  cen 8887  cdom 8888  csdm 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by:  domnsym  9050  gchpwdom  10613
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