Theorem elrpd

Description: Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	elrpd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	elrpd.2	⊢ ( 𝜑 → 0 < 𝐴 )
Assertion	elrpd	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		elrpd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		elrpd.2	⊢ ( 𝜑 → 0 < 𝐴 )
3		elrp	⊢ ( 𝐴 ∈ ℝ+ ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
4	1 2 3	sylanbrc	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )