Theorem elprnq 10905

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 10904	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4041	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3922	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Qcnq 10766  Pcnp 10773

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 3432  df-ss 3907  df-pss 3910  df-np 10895

This theorem is referenced by:  prub  10908  genpv  10913  genpdm  10916  genpss  10918  genpnnp  10919  genpnmax  10921  addclprlem1  10930  addclprlem2  10931  mulclprlem  10933  distrlem4pr  10940  1idpr  10943  psslinpr  10945  prlem934  10947  ltaddpr  10948  ltexprlem2  10951  ltexprlem3  10952  ltexprlem6  10955  ltexprlem7  10956  prlem936  10961  reclem2pr  10962  reclem3pr  10963  reclem4pr  10964