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Theorem soasym 31960
Description: Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
soasym ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Proof of Theorem soasym
StepHypRef Expression
1 sotric 5209 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌𝑌𝑅𝑋)))
2 pm2.46 412 . 2 (¬ (𝑋 = 𝑌𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋)
31, 2syl6bi 243 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1628  wcel 2135   class class class wbr 4800   Or wor 5182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4801  df-po 5183  df-so 5184
This theorem is referenced by:  noresle  32148  noprefixmo  32150  nosupbnd1lem1  32156  nosupbnd1lem4  32159  nosupbnd2lem1  32163  nosupbnd2  32164  sltasym  32175
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