Theorem rprene0d

Description: A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	rprene0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2	1	rpred	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
3	1	rpne0d	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4	2 3	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) )