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Metamath Proof Explorer

Theorem prpssnq

Description: A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996) (Revised by Mario Carneiro, 11-May-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	prpssnq	$⊢ A \in 𝑷 \to A \subset 𝑸$

Proof

Step	Hyp	Ref	Expression
1		elnpi	$⊢ A \in 𝑷 \leftrightarrow ((A \in V \land \emptyset \subset A \land A \subset 𝑸) \land \forall x \in A (\forall y (y <_{𝑸} x \to y \in A) \land \exists y \in A x <_{𝑸} y))$
2		simpl3	$⊢ ((A \in V \land \emptyset \subset A \land A \subset 𝑸) \land \forall x \in A (\forall y (y <_{𝑸} x \to y \in A) \land \exists y \in A x <_{𝑸} y)) \to A \subset 𝑸$
3	1 2	sylbi	$⊢ A \in 𝑷 \to A \subset 𝑸$