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Theorem dfnul2 3458
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 3457 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2348 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3163 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 eqid 2284 . . . . 5  |-  x  =  x
5 pm3.24 852 . . . . 5  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
64, 52th 230 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  _V  /\  -.  x  e.  _V ) )
76con2bii 322 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 262 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2395 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1685   {cab 2270   _Vcvv 2789    \ cdif 3150   (/)c0 3456
This theorem is referenced by:  dfnul3  3459  rab0  3476  iotanul  6268  avril1  20830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-dif 3156  df-nul 3457
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