Theorem elprnq 10879

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 10878	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4050	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3934	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  Qcnq 10740  Pcnp 10747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3919  df-pss 3922  df-np 10869

This theorem is referenced by:  prub  10882  genpv  10887  genpdm  10890  genpss  10892  genpnnp  10893  genpnmax  10895  addclprlem1  10904  addclprlem2  10905  mulclprlem  10907  distrlem4pr  10914  1idpr  10917  psslinpr  10919  prlem934  10921  ltaddpr  10922  ltexprlem2  10925  ltexprlem3  10926  ltexprlem6  10929  ltexprlem7  10930  prlem936  10935  reclem2pr  10936  reclem3pr  10937  reclem4pr  10938