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Mirrors > Home > MPE Home > Th. List > sdomnsym | Structured version Visualization version GIF version |
Description: Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.) |
Ref | Expression |
---|---|
sdomnsym | ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomnen 8224 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
2 | sdomdom 8223 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | sdomdom 8223 | . . 3 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
4 | sbth 8322 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | |
5 | 2, 3, 4 | syl2an 590 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
6 | 1, 5 | mtand 851 | 1 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
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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