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Theorem sdomnsym 8618
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 8514 . 2 (𝐴𝐵 → ¬ 𝐴𝐵)
2 sdomdom 8513 . . 3 (𝐴𝐵𝐴𝐵)
3 sdomdom 8513 . . 3 (𝐵𝐴𝐵𝐴)
4 sbth 8613 . . 3 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
52, 3, 4syl2an 597 . 2 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
61, 5mtand 814 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   class class class wbr 5040  cen 8482  cdom 8483  csdm 8484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-en 8486  df-dom 8487  df-sdom 8488
This theorem is referenced by:  domnsym  8619  gchpwdom  10068
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