Theorem elprnq 10946

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 10945	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4053	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3936	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  Qcnq 10807  Pcnp 10814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-v 3455  df-ss 3921  df-pss 3924  df-np 10936

This theorem is referenced by:  prub  10949  genpv  10954  genpdm  10957  genpss  10959  genpnnp  10960  genpnmax  10962  addclprlem1  10971  addclprlem2  10972  mulclprlem  10974  distrlem4pr  10981  1idpr  10984  psslinpr  10986  prlem934  10988  ltaddpr  10989  ltexprlem2  10992  ltexprlem3  10993  ltexprlem6  10996  ltexprlem7  10997  prlem936  11002  reclem2pr  11003  reclem3pr  11004  reclem4pr  11005