Theorem dfnul2 3365

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 3364	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2206	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3080	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 682	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2139	. . . . 5 |- x = x
6	5	notnoti 634	. . . 4 |- -. -. x = x
7	4, 6	2false 690	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 205	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2254	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   \ cdif 3068   (/)c0 3363

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364

This theorem is referenced by:  dfnul3  3366  rab0  3391  iotanul  5103