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Metamath Proof Explorer

Theorem sdomtr

Description: Strict dominance is transitive. Theorem 21(iii) of Suppes p. 97. (Contributed by NM, 9-Jun-1998)

		Ref	Expression
	Assertion	sdomtr	⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )

Step	Hyp	Ref	Expression
1		sdomdom	⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵 )
2		domsdomtr	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )
3	1 2	sylan	⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶 ) → 𝐴 ≺ 𝐶 )