Theorem sdomnsym

Description: Strict dominance is asymmetric. Theorem 21(ii) of Suppes p. 97. (Contributed by NM, 8-Jun-1998)

	Ref	Expression
Assertion	sdomnsym	⊢ ( 𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sdomnen	⊢ ( 𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵 )
2		sdomdom	⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵 )
3		sdomdom	⊢ ( 𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴 )
4		sbth	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴 ) → 𝐴 ≈ 𝐵 )
5	2 3 4	syl2an	⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴 ) → 𝐴 ≈ 𝐵 )
6	1 5	mtand	⊢ ( 𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴 )