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Metamath Proof Explorer

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	⊢ ( ( 0 ..^ 𝐴 ) ≠ ∅ ↔ 𝐴 ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		fzon0	⊢ ( ( 0 ..^ 𝐴 ) ≠ ∅ ↔ 0 ∈ ( 0 ..^ 𝐴 ) )
2		lbfzo0	⊢ ( 0 ∈ ( 0 ..^ 𝐴 ) ↔ 𝐴 ∈ ℕ )
3	1 2	bitri	⊢ ( ( 0 ..^ 𝐴 ) ≠ ∅ ↔ 𝐴 ∈ ℕ )