Theorem elprnq 11025

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 11024	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4093	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3978	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  Qcnq 10886  Pcnp 10893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-v 3464  df-ss 3963  df-pss 3966  df-np 11015

This theorem is referenced by:  prub  11028  genpv  11033  genpdm  11036  genpss  11038  genpnnp  11039  genpnmax  11041  addclprlem1  11050  addclprlem2  11051  mulclprlem  11053  distrlem4pr  11060  1idpr  11063  psslinpr  11065  prlem934  11067  ltaddpr  11068  ltexprlem2  11071  ltexprlem3  11072  ltexprlem6  11075  ltexprlem7  11076  prlem936  11081  reclem2pr  11082  reclem3pr  11083  reclem4pr  11084