Theorem dfnul2 3470

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 3469	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2274	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3183	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 695	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2207	. . . . 5 |- x = x
6	5	notnoti 646	. . . 4 |- -. -. x = x
7	4, 6	2false 703	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 206	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2322	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   \ cdif 3171   (/)c0 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-nul 3469

This theorem is referenced by:  dfnul3  3471  rab0  3497  iotanul  5266