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

Theorem npex 8865
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 8802 . . 3  |-  Q.  e.  _V
21pwex 4384 . 2  |-  ~P Q.  e.  _V
3 pssss 3444 . . . . 5  |-  ( x 
C.  Q.  ->  x  C_  Q. )
43ad2antlr 709 . . . 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 3417 . . 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 8860 . . 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 3803 . . 3  |-  ~P Q.  =  { x  |  x 
C_  Q. }
85, 6, 73sstr4i 3389 . 2  |-  P.  C_  ~P Q.
92, 8ssexi 4350 1  |-  P.  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322    C. wpss 3323   (/)c0 3630   ~Pcpw 3801   class class class wbr 4214   Q.cnq 8729    <Q cltq 8735   P.cnp 8736
This theorem is referenced by:  enrex  8947  axcnex  9024
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-ni 8751  df-nq 8791  df-np 8860
  Copyright terms: Public domain W3C validator