Theorem soasym 5605

Description: Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

Ref	Expression
soasym	⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Proof of Theorem soasym

Step	Hyp	Ref	Expression
1		sotric 5602	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋)))
2		pm2.46 882	. 2 ⊢ (¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋)
3	1, 2	biimtrdi 253	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   class class class wbr 5123   Or wor 5571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-po 5572  df-so 5573

This theorem is referenced by:  fiinfg  9521  noresle  27679  nosupprefixmo  27682  noinfprefixmo  27683  nosupbnd1lem1  27690  nosupbnd1lem4  27693  nosupbnd2lem1  27697  nosupbnd2  27698  noinfbnd1lem1  27705  noinfbnd1lem4  27708  noinfbnd2lem1  27712  noinfbnd2  27713  sltasym  27730  or2expropbi  47019  prproropf1olem3  47465