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Theorem dfnul2 3772
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 3771 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2521 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3471 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 eqid 2443 . . . . 5  |-  x  =  x
5 pm3.24 882 . . . . 5  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
64, 52th 239 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  _V  /\  -.  x  e.  _V ) )
76con2bii 332 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 271 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2576 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428   _Vcvv 3095    \ cdif 3458   (/)c0 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-dif 3464  df-nul 3771
This theorem is referenced by:  dfnul3  3773  rab0  3792  iotanul  5556  avril1  25043
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