Theorem ne0gt0d

Description: A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ne0gt0d.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	ne0gt0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	ne0gt0d	⊢ ( 𝜑 → 0 < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ne0gt0d.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		ne0gt0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		ne0gt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 ≠ 0 ↔ 0 < 𝐴 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≠ 0 ↔ 0 < 𝐴 ) )
6	3 5	mpbid	⊢ ( 𝜑 → 0 < 𝐴 )