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Theorem npex 9367
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 9304 . . 3  |-  Q.  e.  _V
21pwex 4620 . 2  |-  ~P Q.  e.  _V
3 pssss 3584 . . . . 5  |-  ( x 
C.  Q.  ->  x  C_  Q. )
43ad2antlr 726 . . . 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 3557 . . 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 9362 . . 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 3999 . . 3  |-  ~P Q.  =  { x  |  x 
C_  Q. }
85, 6, 73sstr4i 3528 . 2  |-  P.  C_  ~P Q.
92, 8ssexi 4582 1  |-  P.  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1381    e. wcel 1804   {cab 2428   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461    C. wpss 3462   (/)c0 3770   ~Pcpw 3997   class class class wbr 4437   Q.cnq 9233    <Q cltq 9239   P.cnp 9240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-om 6686  df-ni 9253  df-nq 9293  df-np 9362
This theorem is referenced by:  enrex  9447  axcnex  9527
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