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Theorem intnex 4168
Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
Assertion
Ref Expression
intnex  |-  ( -. 
|^| A  e.  _V  <->  |^| A  =  _V )

Proof of Theorem intnex
StepHypRef Expression
1 intex 4167 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21necon1bbii 2498 . . 3  |-  ( -. 
|^| A  e.  _V  <->  A  =  (/) )
3 inteq 3865 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
4 int0 3876 . . . 4  |-  |^| (/)  =  _V
53, 4syl6eq 2331 . . 3  |-  ( A  =  (/)  ->  |^| A  =  _V )
62, 5sylbi 187 . 2  |-  ( -. 
|^| A  e.  _V  ->  |^| A  =  _V )
7 vprc 4152 . . 3  |-  -.  _V  e.  _V
8 eleq1 2343 . . 3  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
97, 8mtbiri 294 . 2  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
106, 9impbii 180 1  |-  ( -. 
|^| A  e.  _V  <->  |^| A  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   |^|cint 3862
This theorem is referenced by:  intabs  4172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863
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