Theorem elrp

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

	Ref	Expression
Assertion	elrp	$$⊢A\in {ℝ}^{+}\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢x=A\to \left(0<x\leftrightarrow 0<A\right)$$
2		df-rp	$$⊢{ℝ}^{+}=\left\{x\in ℝ|0<x\right\}$$
3	1 2	elrab2	$$⊢A\in {ℝ}^{+}\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$