Theorem dfnul2 3226

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 3225	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2104	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 2927	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 627	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2040	. . . . 5 |- x = x
6	5	notnoti 574	. . . 4 |- -. -. x = x
7	4, 6	2false 617	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 195	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2152	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 97   = wceq 1243   e. wcel 1393   {cab 2026   _Vcvv 2557   \ cdif 2914   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225

This theorem is referenced by:  dfnul3  3227  rab0  3246  iotanul  4882