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Theorem abvor0 3637
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 20 . . . . . 6  |-  ( ph  ->  ph )
2 vex 2951 . . . . . . 7  |-  x  e. 
_V
32a1i 11 . . . . . 6  |-  ( ph  ->  x  e.  _V )
41, 32thd 232 . . . . 5  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
54abbi1dv 2551 . . . 4  |-  ( ph  ->  { x  |  ph }  =  _V )
65con3i 129 . . 3  |-  ( -. 
{ x  |  ph }  =  _V  ->  -. 
ph )
7 id 20 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3624 . . . . . 6  |-  -.  x  e.  (/)
98a1i 11 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 341 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2551 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11syl 16 . 2  |-  ( -. 
{ x  |  ph }  =  _V  ->  { x  |  ph }  =  (/) )
1312orri 366 1  |-  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   (/)c0 3620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-nul 3621
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