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	$$⊢\left(A..^A\right)=\varnothing$$

Proof

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