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Theorem elprnq 8615
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 8614 . . 3  |-  ( A  e.  P.  ->  A  C.  Q. )
21pssssd 3273 . 2  |-  ( A  e.  P.  ->  A  C_ 
Q. )
32sselda 3180 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   Q.cnq 8474   P.cnp 8481
This theorem is referenced by:  prub  8618  genpv  8623  genpdm  8626  genpss  8628  genpnnp  8629  genpnmax  8631  addclprlem1  8640  addclprlem2  8641  mulclprlem  8643  distrlem4pr  8650  1idpr  8653  psslinpr  8655  prlem934  8657  ltaddpr  8658  ltexprlem2  8661  ltexprlem3  8662  ltexprlem6  8665  ltexprlem7  8666  prlem936  8671  reclem2pr  8672  reclem3pr  8673  reclem4pr  8674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-pss 3168  df-np 8605
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