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Theorem elprnq 9386
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 9385 . . 3  |-  ( A  e.  P.  ->  A  C. 
Q. )
21pssssd 3597 . 2  |-  ( A  e.  P.  ->  A  C_ 
Q. )
32sselda 3499 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   Q.cnq 9247   P.cnp 9254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pss 3487  df-np 9376
This theorem is referenced by:  prub  9389  genpv  9394  genpdm  9397  genpss  9399  genpnnp  9400  genpnmax  9402  addclprlem1  9411  addclprlem2  9412  mulclprlem  9414  distrlem4pr  9421  1idpr  9424  psslinpr  9426  prlem934  9428  ltaddpr  9429  ltexprlem2  9432  ltexprlem3  9433  ltexprlem6  9436  ltexprlem7  9437  prlem936  9442  reclem2pr  9443  reclem3pr  9444  reclem4pr  9445
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