Theorem soasym

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

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

Proof

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