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Theorem dfnul2 3740
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 3739 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2480 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3424 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 eqid 2402 . . . . 5  |-  x  =  x
5 pm3.24 883 . . . . 5  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
64, 52th 239 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  _V  /\  -.  x  e.  _V ) )
76con2bii 330 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 271 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2535 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3059    \ cdif 3411   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-dif 3417  df-nul 3739
This theorem is referenced by:  dfnul3  3741  rab0  3760  iotanul  5548  avril1  25588
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