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Metamath Proof Explorer

Theorem cldss

Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006) (Revised by Stefan O'Rear, 22-Feb-2015)

		Ref	Expression
	Hypothesis	iscld.1	$⊢ X = ⋃ J$
	Assertion	cldss	$⊢ S \in Clsd (J) \to S \subseteq X$

Proof

Step	Hyp	Ref	Expression
1		iscld.1	$⊢ X = ⋃ J$
2		cldrcl	$⊢ S \in Clsd (J) \to J \in Top$
3	1	iscld	$⊢ J \in Top \to (S \in Clsd (J) \leftrightarrow (S \subseteq X \land X ∖ S \in J))$
4	3	simprbda	$⊢ (J \in Top \land S \in Clsd (J)) \to S \subseteq X$
5	2 4	mpancom	$⊢ S \in Clsd (J) \to S \subseteq X$