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Metamath Proof Explorer

Theorem sdomen1

Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003)

		Ref	Expression
	Assertion	sdomen1	$⊢ A \approx B \to (A ≺ C \leftrightarrow B ≺ C)$

Step	Hyp	Ref	Expression
1		ensym	$⊢ A \approx B \to B \approx A$
2		ensdomtr	$⊢ (B \approx A \land A ≺ C) \to B ≺ C$
3	1 2	sylan	$⊢ (A \approx B \land A ≺ C) \to B ≺ C$
4		ensdomtr	$⊢ (A \approx B \land B ≺ C) \to A ≺ C$
5	3 4	impbida	$⊢ A \approx B \to (A ≺ C \leftrightarrow B ≺ C)$