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Theorem npex 8655
Description: The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
Assertion
Ref Expression
npex  |-  P.  e.  _V

Proof of Theorem npex
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqex 8592 . . 3  |-  Q.  e.  _V
21pwex 4230 . 2  |-  ~P Q.  e.  _V
3 pssss 3305 . . . . 5  |-  ( x 
C.  Q.  ->  x  C_  Q. )
43ad2antlr 707 . . . 4  |-  ( ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
<Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) )  ->  x  C_  Q. )
54ss2abi 3279 . . 3  |-  { x  |  ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z  <Q 
y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) } 
C_  { x  |  x  C_  Q. }
6 df-np 8650 . . 3  |-  P.  =  { x  |  (
( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
<Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
7 df-pw 3661 . . 3  |-  ~P Q.  =  { x  |  x 
C_  Q. }
85, 6, 73sstr4i 3251 . 2  |-  P.  C_  ~P Q.
92, 8ssexi 4196 1  |-  P.  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578   _Vcvv 2822    C_ wss 3186    C. wpss 3187   (/)c0 3489   ~Pcpw 3659   class class class wbr 4060   Q.cnq 8519    <Q cltq 8525   P.cnp 8526
This theorem is referenced by:  enrex  8737  axcnex  8814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-ni 8541  df-nq 8581  df-np 8650
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