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Theorem elprnq 9158
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 9157 . . 3  |-  ( A  e.  P.  ->  A  C. 
Q. )
21pssssd 3451 . 2  |-  ( A  e.  P.  ->  A  C_ 
Q. )
32sselda 3354 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   Q.cnq 9017   P.cnp 9024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-v 2972  df-in 3333  df-ss 3340  df-pss 3342  df-np 9148
This theorem is referenced by:  prub  9161  genpv  9166  genpdm  9169  genpss  9171  genpnnp  9172  genpnmax  9174  addclprlem1  9183  addclprlem2  9184  mulclprlem  9186  distrlem4pr  9193  1idpr  9196  psslinpr  9198  prlem934  9200  ltaddpr  9201  ltexprlem2  9204  ltexprlem3  9205  ltexprlem6  9208  ltexprlem7  9209  prlem936  9214  reclem2pr  9215  reclem3pr  9216  reclem4pr  9217
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