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Theorem intnex 4558
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 4557 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21necon1bbii 2692 . . 3  |-  ( -. 
|^| A  e.  _V  <->  A  =  (/) )
3 inteq 4229 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
4 int0 4240 . . . 4  |-  |^| (/)  =  _V
53, 4syl6eq 2521 . . 3  |-  ( A  =  (/)  ->  |^| A  =  _V )
62, 5sylbi 200 . 2  |-  ( -. 
|^| A  e.  _V  ->  |^| A  =  _V )
7 vprc 4534 . . 3  |-  -.  _V  e.  _V
8 eleq1 2537 . . 3  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
97, 8mtbiri 310 . 2  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
106, 9impbii 192 1  |-  ( -. 
|^| A  e.  _V  <->  |^| A  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   |^|cint 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-int 4227
This theorem is referenced by:  intabs  4562  relintabex  36258
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