MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnul2 Unicode version

Theorem dfnul2 3459
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 3458 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2349 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3164 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 eqid 2285 . . . . 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 2396 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1625    e. wcel 1686   {cab 2271   _Vcvv 2790    \ cdif 3151   (/)c0 3457
This theorem is referenced by:  dfnul3  3460  rab0  3477  iotanul  5236  avril1  20838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-dif 3157  df-nul 3458
  Copyright terms: Public domain W3C validator