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Theorem elprnq 9810
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 ((𝐴P𝐵𝐴) → 𝐵Q)

Proof of Theorem elprnq
StepHypRef Expression
1 prpssnq 9809 . . 3 (𝐴P𝐴Q)
21pssssd 3702 . 2 (𝐴P𝐴Q)
32sselda 3601 1 ((𝐴P𝐵𝐴) → 𝐵Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  Qcnq 9671  Pcnp 9678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-v 3200  df-in 3579  df-ss 3586  df-pss 3588  df-np 9800
This theorem is referenced by:  prub  9813  genpv  9818  genpdm  9821  genpss  9823  genpnnp  9824  genpnmax  9826  addclprlem1  9835  addclprlem2  9836  mulclprlem  9838  distrlem4pr  9845  1idpr  9848  psslinpr  9850  prlem934  9852  ltaddpr  9853  ltexprlem2  9856  ltexprlem3  9857  ltexprlem6  9860  ltexprlem7  9861  prlem936  9866  reclem2pr  9867  reclem3pr  9868  reclem4pr  9869
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