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Mirrors > Home > MPE Home > Th. List > soasym | Structured version Visualization version GIF version |
Description: Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
Ref | Expression |
---|---|
soasym | ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sotric 5625 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋))) | |
2 | pm2.46 882 | . 2 ⊢ (¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋) | |
3 | 1, 2 | biimtrdi 253 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 Or wor 5595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-po 5596 df-so 5597 |
This theorem is referenced by: fiinfg 9536 noresle 27756 nosupprefixmo 27759 noinfprefixmo 27760 nosupbnd1lem1 27767 nosupbnd1lem4 27770 nosupbnd2lem1 27774 nosupbnd2 27775 noinfbnd1lem1 27782 noinfbnd1lem4 27785 noinfbnd2lem1 27789 noinfbnd2 27790 sltasym 27807 or2expropbi 46983 prproropf1olem3 47429 |
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