Theorem elprnq 11029

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.)

Assertion

Ref	Expression
elprnq	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Proof of Theorem elprnq

Step	Hyp	Ref	Expression
1		prpssnq 11028	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4110	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3995	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Qcnq 10890  Pcnp 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-pss 3983  df-np 11019

This theorem is referenced by:  prub  11032  genpv  11037  genpdm  11040  genpss  11042  genpnnp  11043  genpnmax  11045  addclprlem1  11054  addclprlem2  11055  mulclprlem  11057  distrlem4pr  11064  1idpr  11067  psslinpr  11069  prlem934  11071  ltaddpr  11072  ltexprlem2  11075  ltexprlem3  11076  ltexprlem6  11079  ltexprlem7  11080  prlem936  11085  reclem2pr  11086  reclem3pr  11087  reclem4pr  11088