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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 5622 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋))) | |
| 2 | pm2.46 883 | . 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 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 Or wor 5591 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-po 5592 df-so 5593 |
| This theorem is referenced by: fiinfg 9539 noresle 27742 nosupprefixmo 27745 noinfprefixmo 27746 nosupbnd1lem1 27753 nosupbnd1lem4 27756 nosupbnd2lem1 27760 nosupbnd2 27761 noinfbnd1lem1 27768 noinfbnd1lem4 27771 noinfbnd2lem1 27775 noinfbnd2 27776 sltasym 27793 or2expropbi 47046 prproropf1olem3 47492 |
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