Theorem soasym 5525

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 5522	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋)))
2		pm2.46 879	. 2 ⊢ (¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋) → ¬ 𝑌𝑅𝑋)
3	1, 2	syl6bi 252	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   class class class wbr 5070   Or wor 5493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-po 5494  df-so 5495

This theorem is referenced by:  fiinfg  9188  noresle  33827  nosupprefixmo  33830  noinfprefixmo  33831  nosupbnd1lem1  33838  nosupbnd1lem4  33841  nosupbnd2lem1  33845  nosupbnd2  33846  noinfbnd1lem1  33853  noinfbnd1lem4  33856  noinfbnd2lem1  33860  noinfbnd2  33861  sltasym  33878  or2expropbi  44415  prproropf1olem3  44845