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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 8138 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
2 | sdomdom 8137 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | sdomdom 8137 | . . 3 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
4 | sbth 8236 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | |
5 | 2, 3, 4 | syl2an 583 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
6 | 1, 5 | mtand 817 | 1 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 class class class wbr 4786 ≈ cen 8106 ≼ cdom 8107 ≺ csdm 8108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-en 8110 df-dom 8111 df-sdom 8112 |
This theorem is referenced by: domnsym 8242 gchpwdom 9694 |
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