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Metamath Proof Explorer

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	$⊢ (A \in 𝑷 \land B \in A) \to B \in 𝑸$

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	$⊢ (A \in 𝑷 \land B \in A) \to B \in 𝑸$