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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 5494 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋))) | |
2 | pm2.46 876 | . 2 ⊢ (¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋) | |
3 | 1, 2 | syl6bi 254 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 Or wor 5466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-po 5467 df-so 5468 |
This theorem is referenced by: fiinfg 8951 noresle 33097 noprefixmo 33099 nosupbnd1lem1 33105 nosupbnd1lem4 33108 nosupbnd2lem1 33112 nosupbnd2 33113 sltasym 33124 or2expropbi 43146 prproropf1olem3 43544 |
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