Theorem elprnq 9813

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

Step	Hyp	Ref	Expression
1		prpssnq 9812	. . 3 |- ( A e. P. -> A C. Q. )
2	1	pssssd 3704	. 2 |- ( A e. P. -> A C_ Q. )
3	2	sselda 3603	1 |- ( ( A e. P. /\ B e. A ) -> B e. Q. )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   Q.cnq 9674   P.cnp 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pss 3590  df-np 9803

This theorem is referenced by:  prub  9816  genpv  9821  genpdm  9824  genpss  9826  genpnnp  9827  genpnmax  9829  addclprlem1  9838  addclprlem2  9839  mulclprlem  9841  distrlem4pr  9848  1idpr  9851  psslinpr  9853  prlem934  9855  ltaddpr  9856  ltexprlem2  9859  ltexprlem3  9860  ltexprlem6  9863  ltexprlem7  9864  prlem936  9869  reclem2pr  9870  reclem3pr  9871  reclem4pr  9872