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Metamath Proof Explorer

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