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

Theorem npex 9418
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 9355 . . 3  |-  Q.  e.  _V
21pwex 4607 . 2  |-  ~P Q.  e.  _V
3 pssss 3560 . . . . 5  |-  ( x 
C.  Q.  ->  x  C_  Q. )
43ad2antlr 731 . . . 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 3533 . . 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 9413 . . 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 3983 . . 3  |-  ~P Q.  =  { x  |  x 
C_  Q. }
85, 6, 73sstr4i 3503 . 2  |-  P.  C_  ~P Q.
92, 8ssexi 4569 1  |-  P.  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435    e. wcel 1872   {cab 2407   A.wral 2771   E.wrex 2772   _Vcvv 3080    C_ wss 3436    C. wpss 3437   (/)c0 3761   ~Pcpw 3981   class class class wbr 4423   Q.cnq 9284    <Q cltq 9290   P.cnp 9291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6707  df-ni 9304  df-nq 9344  df-np 9413
This theorem is referenced by:  enrex  9498  axcnex  9578
  Copyright terms: Public domain W3C validator