Theorem elrp

Description: Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007)

	Ref	Expression
Assertion	elrp	⊢ ( 𝐴 ∈ ℝ+ ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑥 = 𝐴 → ( 0 < 𝑥 ↔ 0 < 𝐴 ) )
2		df-rp	⊢ ℝ+ = { 𝑥 ∈ ℝ ∣ 0 < 𝑥 }
3	1 2	elrab2	⊢ ( 𝐴 ∈ ℝ+ ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )