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Theorem dfnul2 3271
Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
Assertion
Ref Expression
dfnul2  |-  (/)  =  {
x  |  -.  x  =  x }

Proof of Theorem dfnul2
StepHypRef Expression
1 df-nul 3270 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2149 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 2993 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 pm3.24 660 . . . 4  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
5 eqid 2083 . . . . 5  |-  x  =  x
65notnoti 607 . . . 4  |-  -.  -.  x  =  x
74, 62false 650 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 204 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2197 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2069   _Vcvv 2612    \ cdif 2981   (/)c0 3269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-nul 3270
This theorem is referenced by:  dfnul3  3272  rab0  3294  iotanul  4949
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