Theorem elprnq 11103

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 11102	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4030	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3913	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Qcnq 10964  Pcnp 10971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-pss 4017  df-np 11093

This theorem is used by:  prub  11106  genpv  11111  genpdm  11114  genpss  11116  genpnnp  11117  genpnmax  11119  addclprlem1  11128  addclprlem2  11129  mulclprlem  11131  distrlem4pr  11138  1idpr  11141  psslinpr  11143  prlem934  11145  ltaddpr  11146  ltexprlem2  11149  ltexprlem3  11150  ltexprlem6  11153  ltexprlem7  11154  prlem936  11159  reclem2pr  11160  reclem3pr  11161  reclem4pr  11162