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Metamath Proof Explorer

Theorem sncldre

Description: A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019)

		Ref	Expression
	Assertion	sncldre	$⊢ A \in ℝ \to \{A\} \in Clsd (topGen (ran (.)))$

Step	Hyp	Ref	Expression
1		rehaus	$⊢ topGen (ran (.)) \in Haus$
2		uniretop	$⊢ ℝ = ⋃ topGen (ran (.))$
3	2	sncld	$⊢ (topGen (ran (.)) \in Haus \land A \in ℝ) \to \{A\} \in Clsd (topGen (ran (.)))$
4	1 3	mpan	$⊢ A \in ℝ \to \{A\} \in Clsd (topGen (ran (.)))$