Theorem repos

Description: Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007)

	Ref	Expression
Assertion	repos	⊢ ( 𝐴 ∈ ( 0 (,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )

Proof

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