Theorem elprnq 10413

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 10412	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4074	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3967	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  Qcnq 10274  Pcnp 10281

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-in 3943  df-ss 3952  df-pss 3954  df-np 10403

This theorem is referenced by:  prub  10416  genpv  10421  genpdm  10424  genpss  10426  genpnnp  10427  genpnmax  10429  addclprlem1  10438  addclprlem2  10439  mulclprlem  10441  distrlem4pr  10448  1idpr  10451  psslinpr  10453  prlem934  10455  ltaddpr  10456  ltexprlem2  10459  ltexprlem3  10460  ltexprlem6  10463  ltexprlem7  10464  prlem936  10469  reclem2pr  10470  reclem3pr  10471  reclem4pr  10472