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Theorem dfnul2 3737
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 3736 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2529 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 3436 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 eqid 2451 . . . . 5  |-  x  =  x
5 pm3.24 877 . . . . 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 2584 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2436   _Vcvv 3068    \ cdif 3423   (/)c0 3735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-dif 3429  df-nul 3736
This theorem is referenced by:  dfnul3  3738  rab0  3756  iotanul  5494  avril1  23791
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