Theorem elprnq 10914

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 10913	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4054	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3935	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Qcnq 10775  Pcnp 10782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3444  df-ss 3920  df-pss 3923  df-np 10904

This theorem is referenced by:  prub  10917  genpv  10922  genpdm  10925  genpss  10927  genpnnp  10928  genpnmax  10930  addclprlem1  10939  addclprlem2  10940  mulclprlem  10942  distrlem4pr  10949  1idpr  10952  psslinpr  10954  prlem934  10956  ltaddpr  10957  ltexprlem2  10960  ltexprlem3  10961  ltexprlem6  10964  ltexprlem7  10965  prlem936  10970  reclem2pr  10971  reclem3pr  10972  reclem4pr  10973