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Theorem prpssnq 9357
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 9355 . 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 999 . 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 195 1  |-  ( A  e.  P.  ->  A  C. 
Q. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971   A.wal 1396    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    C. wpss 3462   (/)c0 3783   class class class wbr 4439   Q.cnq 9219    <Q cltq 9225   P.cnp 9226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-pss 3477  df-np 9348
This theorem is referenced by:  elprnq  9358  npomex  9363  genpnnp  9372  prlem934  9400  ltexprlem2  9404  reclem2pr  9415  suplem1pr  9419  wuncn  9536
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