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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 5470 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋))) | |
2 | pm2.46 882 | . 2 ⊢ (¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋) | |
3 | 1, 2 | syl6bi 256 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 Or wor 5441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-v 3400 df-un 3848 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-po 5442 df-so 5443 |
This theorem is referenced by: fiinfg 9036 noresle 33543 nosupprefixmo 33546 noinfprefixmo 33547 nosupbnd1lem1 33554 nosupbnd1lem4 33557 nosupbnd2lem1 33561 nosupbnd2 33562 noinfbnd1lem1 33569 noinfbnd1lem4 33572 noinfbnd2lem1 33576 noinfbnd2 33577 sltasym 33594 or2expropbi 44087 prproropf1olem3 44511 |
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