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Metamath Proof Explorer

Theorem mrcsncl

Description: The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015)

		Ref	Expression
	Hypothesis	mrcfval.f	$⊢ F = mrCls (C)$
	Assertion	mrcsncl	$⊢ (C \in Moore (X) \land U \in X) \to F (\{U\}) \in C$

Step	Hyp	Ref	Expression
1		mrcfval.f	$⊢ F = mrCls (C)$
2		snssi	$⊢ U \in X \to \{U\} \subseteq X$
3	1	mrccl	$⊢ (C \in Moore (X) \land \{U\} \subseteq X) \to F (\{U\}) \in C$
4	2 3	sylan2	$⊢ (C \in Moore (X) \land U \in X) \to F (\{U\}) \in C$