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Theorem dfnul2 3253
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 3252 . . . 4  |-  (/)  =  ( _V  \  _V )
21eleq2i 2120 . . 3  |-  ( x  e.  (/)  <->  x  e.  ( _V  \  _V ) )
3 eldif 2954 . . 3  |-  ( x  e.  ( _V  \  _V )  <->  ( x  e. 
_V  /\  -.  x  e.  _V ) )
4 pm3.24 637 . . . 4  |-  -.  (
x  e.  _V  /\  -.  x  e.  _V )
5 eqid 2056 . . . . 5  |-  x  =  x
65notnoti 584 . . . 4  |-  -.  -.  x  =  x
74, 62false 627 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  _V ) 
<->  -.  x  =  x )
82, 3, 73bitri 199 . 2  |-  ( x  e.  (/)  <->  -.  x  =  x )
98abbi2i 2168 1  |-  (/)  =  {
x  |  -.  x  =  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    = wceq 1259    e. wcel 1409   {cab 2042   _Vcvv 2574    \ cdif 2941   (/)c0 3251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-nul 3252
This theorem is referenced by:  dfnul3  3254  rab0  3273  iotanul  4909
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