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Theorem dfnul2 3335
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 3334 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2184 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3050 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 pm3.24 667 . . . 4  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
5 eqid 2117 . . . . 5  |-  x  =  x
65notnoti 619 . . . 4  |-  -.  -.  x  =  x
74, 62false 675 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 205 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2232 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660    \ cdif 3038   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by:  dfnul3  3336  rab0  3361  iotanul  5073
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