Theorem elprnq 10986

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 10985	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4098	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3983	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Qcnq 10847  Pcnp 10854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-pss 3968  df-np 10976

This theorem is referenced by:  prub  10989  genpv  10994  genpdm  10997  genpss  10999  genpnnp  11000  genpnmax  11002  addclprlem1  11011  addclprlem2  11012  mulclprlem  11014  distrlem4pr  11021  1idpr  11024  psslinpr  11026  prlem934  11028  ltaddpr  11029  ltexprlem2  11032  ltexprlem3  11033  ltexprlem6  11036  ltexprlem7  11037  prlem936  11042  reclem2pr  11043  reclem3pr  11044  reclem4pr  11045