Theorem rpregt0d

Description: A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
Assertion	rpregt0d	$$⊢\phi \to \left(A\in ℝ\wedge 0<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2	1	rpred	$$⊢\phi \to A\in ℝ$$
3	1	rpgt0d	$$⊢\phi \to 0<A$$
4	2 3	jca	$$⊢\phi \to \left(A\in ℝ\wedge 0<A\right)$$