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Metamath Proof Explorer

Theorem mress

Description: A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015)

		Ref	Expression
	Assertion	mress	$⊢ (C \in Moore (X) \land S \in C) \to S \subseteq X$

Step	Hyp	Ref	Expression
1		mresspw	$⊢ C \in Moore (X) \to C \subseteq 𝒫 X$
2	1	sselda	$⊢ (C \in Moore (X) \land S \in C) \to S \in 𝒫 X$
3	2	elpwid	$⊢ (C \in Moore (X) \land S \in C) \to S \subseteq X$