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Metamath Proof Explorer

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 )