Theorem repos

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

	Ref	Expression
Assertion	repos	$$⊢A\in \left(0,\mathrm{+\infty }\right)\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢x=A\to \left(0<x\leftrightarrow 0<A\right)$$
2		ioopos	$$⊢\left(0,\mathrm{+\infty }\right)=\left\{x\in ℝ|0<x\right\}$$
3	1 2	elrab2	$$⊢A\in \left(0,\mathrm{+\infty }\right)\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$