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Theorem soasym 5534
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 5531 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌𝑌𝑅𝑋)))
2 pm2.46 880 . 2 (¬ (𝑋 = 𝑌𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋)
31, 2syl6bi 252 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844   = wceq 1539  wcel 2106   class class class wbr 5074   Or wor 5502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-po 5503  df-so 5504
This theorem is referenced by:  fiinfg  9258  noresle  33900  nosupprefixmo  33903  noinfprefixmo  33904  nosupbnd1lem1  33911  nosupbnd1lem4  33914  nosupbnd2lem1  33918  nosupbnd2  33919  noinfbnd1lem1  33926  noinfbnd1lem4  33929  noinfbnd2lem1  33933  noinfbnd2  33934  sltasym  33951  or2expropbi  44528  prproropf1olem3  44957
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