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Metamath Proof Explorer

Theorem fzo0

Description: Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015) (Revised by Mario Carneiro, 29-Sep-2015)

		Ref	Expression
	Assertion	fzo0	$⊢ (A ..^A) = \emptyset$

Step	Hyp	Ref	Expression
1		fzonel	$⊢ \neg A \in (A ..^A)$
2		fzon0	$⊢ (A ..^A) \neq \emptyset \leftrightarrow A \in (A ..^A)$
3	2	necon1bbii	$⊢ \neg A \in (A ..^A) \leftrightarrow (A ..^A) = \emptyset$
4	1 3	mpbi	$⊢ (A ..^A) = \emptyset$