Theorem elprnq

Description: A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996) (Revised by Mario Carneiro, 11-May-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	elprnq	$$⊢\left(A\in 𝑷\wedge B\in A\right)\to B\in 𝑸$$

Proof

Step	Hyp	Ref	Expression
1		prpssnq	$$⊢A\in 𝑷\to A\subset 𝑸$$
2	1	pssssd	$$⊢A\in 𝑷\to A\subseteq 𝑸$$
3	2	sselda	$$⊢\left(A\in 𝑷\wedge B\in A\right)\to B\in 𝑸$$