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 ∈ P ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ Q )

Proof

Step	Hyp	Ref	Expression
1		prpssnq	⊢ ( 𝐴 ∈ P → 𝐴 ⊊ Q )
2	1	pssssd	⊢ ( 𝐴 ∈ P → 𝐴 ⊆ Q )
3	2	sselda	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ Q )