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Metamath Proof Explorer

Theorem pnrmnrm

Description: A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015)

		Ref	Expression
	Assertion	pnrmnrm	$⊢ J \in PNrm \to J \in Nrm$

Step	Hyp	Ref	Expression
1		ispnrm	$⊢ J \in PNrm \leftrightarrow (J \in Nrm \land Clsd (J) \subseteq ran (x \in (J^{ℕ}) ⟼ ⋂ ran x))$
2	1	simplbi	$⊢ J \in PNrm \to J \in Nrm$