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Theorem intnex 4386
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 4385 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21necon1bbii 2662 . . 3  |-  ( -. 
|^| A  e.  _V  <->  A  =  (/) )
3 inteq 4077 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
4 int0 4088 . . . 4  |-  |^| (/)  =  _V
53, 4syl6eq 2490 . . 3  |-  ( A  =  (/)  ->  |^| A  =  _V )
62, 5sylbi 189 . 2  |-  ( -. 
|^| A  e.  _V  ->  |^| A  =  _V )
7 vprc 4370 . . 3  |-  -.  _V  e.  _V
8 eleq1 2502 . . 3  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
97, 8mtbiri 296 . 2  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
106, 9impbii 182 1  |-  ( -. 
|^| A  e.  _V  <->  |^| A  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    = wceq 1653    e. wcel 1727   _Vcvv 2962   (/)c0 3613   |^|cint 4074
This theorem is referenced by:  intabs  4390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-v 2964  df-dif 3309  df-in 3313  df-ss 3320  df-nul 3614  df-int 4075
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