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Theorem elprnq 10014
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 10013 . . 3 (𝐴P𝐴Q)
21pssssd 3852 . 2 (𝐴P𝐴Q)
32sselda 3750 1 ((𝐴P𝐵𝐴) → 𝐵Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  Qcnq 9875  Pcnp 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-pss 3737  df-np 10004
This theorem is referenced by:  prub  10017  genpv  10022  genpdm  10025  genpss  10027  genpnnp  10028  genpnmax  10030  addclprlem1  10039  addclprlem2  10040  mulclprlem  10042  distrlem4pr  10049  1idpr  10052  psslinpr  10054  prlem934  10056  ltaddpr  10057  ltexprlem2  10060  ltexprlem3  10061  ltexprlem6  10064  ltexprlem7  10065  prlem936  10070  reclem2pr  10071  reclem3pr  10072  reclem4pr  10073
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