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Theorem sdomnsym 8327
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 8224 . 2 (𝐴𝐵 → ¬ 𝐴𝐵)
2 sdomdom 8223 . . 3 (𝐴𝐵𝐴𝐵)
3 sdomdom 8223 . . 3 (𝐵𝐴𝐵𝐴)
4 sbth 8322 . . 3 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
52, 3, 4syl2an 590 . 2 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
61, 5mtand 851 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   class class class wbr 4843  cen 8192  cdom 8193  csdm 8194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-en 8196  df-dom 8197  df-sdom 8198
This theorem is referenced by:  domnsym  8328  gchpwdom  9780
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