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Theorem abvor0 3796
Description: The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)
Assertion
Ref Expression
abvor0  |-  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) )
Distinct variable group:    ph, x

Proof of Theorem abvor0
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( ph  ->  ph )
2 vex 3109 . . . . . . 7  |-  x  e. 
_V
32a1i 11 . . . . . 6  |-  ( ph  ->  x  e.  _V )
41, 32thd 240 . . . . 5  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
54abbi1dv 2598 . . . 4  |-  ( ph  ->  { x  |  ph }  =  _V )
65con3i 135 . . 3  |-  ( -. 
{ x  |  ph }  =  _V  ->  -. 
ph )
7 id 22 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3782 . . . . . 6  |-  -.  x  e.  (/)
98a1i 11 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 351 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2598 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11syl 16 . 2  |-  ( -. 
{ x  |  ph }  =  _V  ->  { x  |  ph }  =  (/) )
1312orri 376 1  |-  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1374    e. wcel 1762   {cab 2445   _Vcvv 3106   (/)c0 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-dif 3472  df-nul 3779
This theorem is referenced by: (None)
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