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

Step	Hyp	Ref	Expression
1		df-nul 3252	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2145	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 2982	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 659	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2081	. . . . 5 |- x = x
6	5	notnoti 606	. . . 4 |- -. -. x = x
7	4, 6	2false 649	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 204	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2193	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   \ cdif 2970   (/)c0 3251

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by:  dfnul3  3254  rab0  3273  iotanul  4902