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Theorem abvor0 3756
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 3074 . . . . . . 7  |-  x  e. 
_V
32a1i 11 . . . . . 6  |-  ( ph  ->  x  e.  _V )
41, 32thd 240 . . . . 5  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
54abbi1dv 2589 . . . 4  |-  ( ph  ->  { x  |  ph }  =  _V )
65con3i 135 . . 3  |-  ( -. 
{ x  |  ph }  =  _V  ->  -. 
ph )
7 id 22 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3742 . . . . . 6  |-  -.  x  e.  (/)
98a1i 11 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 351 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2589 . . 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 1370    e. wcel 1758   {cab 2436   _Vcvv 3071   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-dif 3432  df-nul 3739
This theorem is referenced by: (None)
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