Theorem dfnul2 3271

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

Step	Hyp	Ref	Expression
1		df-nul 3270	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2149	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 2993	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 660	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2083	. . . . 5 |- x = x
6	5	notnoti 607	. . . 4 |- -. -. x = x
7	4, 6	2false 650	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 204	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2197	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1285   e. wcel 1434   {cab 2069   _Vcvv 2612   \ cdif 2981   (/)c0 3269

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-nul 3270

This theorem is referenced by:  dfnul3  3272  rab0  3294  iotanul  4949