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Theorem intnex 4325
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 4324 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21necon1bbii 2627 . . 3  |-  ( -. 
|^| A  e.  _V  <->  A  =  (/) )
3 inteq 4021 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
4 int0 4032 . . . 4  |-  |^| (/)  =  _V
53, 4syl6eq 2460 . . 3  |-  ( A  =  (/)  ->  |^| A  =  _V )
62, 5sylbi 188 . 2  |-  ( -. 
|^| A  e.  _V  ->  |^| A  =  _V )
7 vprc 4309 . . 3  |-  -.  _V  e.  _V
8 eleq1 2472 . . 3  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
97, 8mtbiri 295 . 2  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
106, 9impbii 181 1  |-  ( -. 
|^| A  e.  _V  <->  |^| A  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2924   (/)c0 3596   |^|cint 4018
This theorem is referenced by:  intabs  4329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-v 2926  df-dif 3291  df-in 3295  df-ss 3302  df-nul 3597  df-int 4019
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