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Metamath Proof Explorer

Theorem sdomn2lp

Description: Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008)

		Ref	Expression
	Assertion	sdomn2lp	⊢ ¬ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴 )

Step	Hyp	Ref	Expression
1		sdomirr	⊢ ¬ 𝐴 ≺ 𝐴
2		sdomtr	⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴 ) → 𝐴 ≺ 𝐴 )
3	1 2	mto	⊢ ¬ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴 )