MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prpssnq Structured version   Unicode version

Theorem prpssnq 8869
Description: A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prpssnq  |-  ( A  e.  P.  ->  A  C.  Q. )

Proof of Theorem prpssnq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnpi 8867 . 2  |-  ( A  e.  P.  <->  ( ( A  e.  _V  /\  (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y 
<Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
2 simpl3 963 . 2  |-  ( ( ( A  e.  _V  /\  (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) )  ->  A  C.  Q. )
31, 2sylbi 189 1  |-  ( A  e.  P.  ->  A  C.  Q. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C. wpss 3323   (/)c0 3630   class class class wbr 4214   Q.cnq 8729    <Q cltq 8735   P.cnp 8736
This theorem is referenced by:  elprnq  8870  npomex  8875  genpnnp  8884  prlem934  8912  ltexprlem2  8916  reclem2pr  8927  suplem1pr  8931  wuncn  9047
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 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-pss 3338  df-np 8860
  Copyright terms: Public domain W3C validator