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Theorem abvor0 3786
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 23 . . . . . 6  |-  ( ph  ->  ph )
2 vex 3090 . . . . . . 7  |-  x  e. 
_V
32a1i 11 . . . . . 6  |-  ( ph  ->  x  e.  _V )
41, 32thd 243 . . . . 5  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
54abbi1dv 2567 . . . 4  |-  ( ph  ->  { x  |  ph }  =  _V )
65con3i 140 . . 3  |-  ( -. 
{ x  |  ph }  =  _V  ->  -. 
ph )
7 id 23 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3771 . . . . . 6  |-  -.  x  e.  (/)
98a1i 11 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 352 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2567 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11syl 17 . 2  |-  ( -. 
{ x  |  ph }  =  _V  ->  { x  |  ph }  =  (/) )
1312orri 377 1  |-  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    = wceq 1437    e. wcel 1870   {cab 2414   _Vcvv 3087   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-dif 3445  df-nul 3768
This theorem is referenced by: (None)
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