Theorem elrpd

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

	Ref	Expression
Hypotheses	elrpd.1	$$⊢\phi \to A\in ℝ$$
	elrpd.2	$$⊢\phi \to 0<A$$
Assertion	elrpd	$$⊢\phi \to A\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		elrpd.1	$$⊢\phi \to A\in ℝ$$
2		elrpd.2	$$⊢\phi \to 0<A$$
3		elrp	$$⊢A\in {ℝ}^{+}\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$
4	1 2 3	sylanbrc	$$⊢\phi \to A\in {ℝ}^{+}$$