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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 9004 | . 2 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
2 | sdomdom 9003 | . . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | sdomdom 9003 | . . 3 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
4 | sbth 9123 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | |
5 | 2, 3, 4 | syl2an 594 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
6 | 1, 5 | mtand 814 | 1 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 class class class wbr 5145 ≈ cen 8963 ≼ cdom 8964 ≺ csdm 8965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-en 8967 df-dom 8968 df-sdom 8969 |
This theorem is referenced by: domnsym 9129 gchpwdom 10704 |
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