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Theorem elprnq 8869
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  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )

Proof of Theorem elprnq
StepHypRef Expression
1 prpssnq 8868 . . 3  |-  ( A  e.  P.  ->  A  C.  Q. )
21pssssd 3445 . 2  |-  ( A  e.  P.  ->  A  C_ 
Q. )
32sselda 3349 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   Q.cnq 8728   P.cnp 8735
This theorem is referenced by:  prub  8872  genpv  8877  genpdm  8880  genpss  8882  genpnnp  8883  genpnmax  8885  addclprlem1  8894  addclprlem2  8895  mulclprlem  8897  distrlem4pr  8904  1idpr  8907  psslinpr  8909  prlem934  8911  ltaddpr  8912  ltexprlem2  8915  ltexprlem3  8916  ltexprlem6  8919  ltexprlem7  8920  prlem936  8925  reclem2pr  8926  reclem3pr  8927  reclem4pr  8928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-in 3328  df-ss 3335  df-pss 3337  df-np 8859
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