Theorem dfnul2 3370

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 3369	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2207	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3085	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 683	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2140	. . . . 5 |- x = x
6	5	notnoti 635	. . . 4 |- -. -. x = x
7	4, 6	2false 691	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 205	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2255	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689   \ cdif 3073   (/)c0 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-nul 3369

This theorem is referenced by:  dfnul3  3371  rab0  3396  iotanul  5111