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Theorem sdomnsym 8045
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 7944 . 2 (𝐴𝐵 → ¬ 𝐴𝐵)
2 sdomdom 7943 . . 3 (𝐴𝐵𝐴𝐵)
3 sdomdom 7943 . . 3 (𝐵𝐴𝐵𝐴)
4 sbth 8040 . . 3 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
52, 3, 4syl2an 494 . 2 ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
61, 5mtand 690 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   class class class wbr 4623  cen 7912  cdom 7913  csdm 7914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-en 7916  df-dom 7917  df-sdom 7918
This theorem is referenced by:  domnsym  8046  gchpwdom  9452
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