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Metamath Proof Explorer

Theorem domen1

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

		Ref	Expression
	Assertion	domen1	$⊢ A \approx B \to (A ≼ C \leftrightarrow B ≼ C)$

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