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Theorem abvor0 3485
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 19 . . . . . 6  |-  ( ph  ->  ph )
2 vex 2804 . . . . . . 7  |-  x  e. 
_V
32a1i 10 . . . . . 6  |-  ( ph  ->  x  e.  _V )
41, 32thd 231 . . . . 5  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
54abbi1dv 2412 . . . 4  |-  ( ph  ->  { x  |  ph }  =  _V )
65con3i 127 . . 3  |-  ( -. 
{ x  |  ph }  =  _V  ->  -. 
ph )
7 id 19 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3472 . . . . . 6  |-  -.  x  e.  (/)
98a1i 10 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 340 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2412 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11syl 15 . 2  |-  ( -. 
{ x  |  ph }  =  _V  ->  { x  |  ph }  =  (/) )
1312orri 365 1  |-  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-nul 3469
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