Theorem fzo0n0

Description: A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015) (Revised by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	fzo0n0	$$⊢\left(0..^A\right)\ne \varnothing \leftrightarrow A\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		fzon0	$$⊢\left(0..^A\right)\ne \varnothing \leftrightarrow 0\in \left(0..^A\right)$$
2		lbfzo0	$$⊢0\in \left(0..^A\right)\leftrightarrow A\in \mathbb{N}$$
3	1 2	bitri	$$⊢\left(0..^A\right)\ne \varnothing \leftrightarrow A\in \mathbb{N}$$