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Mirrors > Home > MPE Home > Th. List > sdomn2lp | Structured version Visualization version GIF version |
Description: Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.) |
Ref | Expression |
---|---|
sdomn2lp | ⊢ ¬ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomirr 9153 | . 2 ⊢ ¬ 𝐴 ≺ 𝐴 | |
2 | sdomtr 9154 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≺ 𝐴) | |
3 | 1, 2 | mto 197 | 1 ⊢ ¬ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 class class class wbr 5148 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: (None) |
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