Theorem elprnq 10415

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 10414	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
2	1	pssssd 4076	. 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
3	2	sselda 3969	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  Qcnq 10276  Pcnp 10283

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 2795

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-v 3498  df-in 3945  df-ss 3954  df-pss 3956  df-np 10405

This theorem is referenced by:  prub  10418  genpv  10423  genpdm  10426  genpss  10428  genpnnp  10429  genpnmax  10431  addclprlem1  10440  addclprlem2  10441  mulclprlem  10443  distrlem4pr  10450  1idpr  10453  psslinpr  10455  prlem934  10457  ltaddpr  10458  ltexprlem2  10461  ltexprlem3  10462  ltexprlem6  10465  ltexprlem7  10466  prlem936  10471  reclem2pr  10472  reclem3pr  10473  reclem4pr  10474