Theorem soasym 5625

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 5622	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 ↔ ¬ (𝑋 = 𝑌 ∨ 𝑌𝑅𝑋)))
2		pm2.46 883	. 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 848   = wceq 1540   ∈ wcel 2108   class class class wbr 5143   Or wor 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-po 5592  df-so 5593

This theorem is referenced by:  fiinfg  9539  noresle  27742  nosupprefixmo  27745  noinfprefixmo  27746  nosupbnd1lem1  27753  nosupbnd1lem4  27756  nosupbnd2lem1  27760  nosupbnd2  27761  noinfbnd1lem1  27768  noinfbnd1lem4  27771  noinfbnd2lem1  27775  noinfbnd2  27776  sltasym  27793  or2expropbi  47046  prproropf1olem3  47492