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Theorem intnex 4270
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 4269 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21necon1bbii 2581 . . 3  |-  ( -. 
|^| A  e.  _V  <->  A  =  (/) )
3 inteq 3967 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
4 int0 3978 . . . 4  |-  |^| (/)  =  _V
53, 4syl6eq 2414 . . 3  |-  ( A  =  (/)  ->  |^| A  =  _V )
62, 5sylbi 187 . 2  |-  ( -. 
|^| A  e.  _V  ->  |^| A  =  _V )
7 vprc 4254 . . 3  |-  -.  _V  e.  _V
8 eleq1 2426 . . 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 1647    e. wcel 1715   _Vcvv 2873   (/)c0 3543   |^|cint 3964
This theorem is referenced by:  intabs  4274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-v 2875  df-dif 3241  df-in 3245  df-ss 3252  df-nul 3544  df-int 3965
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