Theorem dfnul2 3416

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 3415	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2237	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3130	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 688	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2170	. . . . 5 |- x = x
6	5	notnoti 640	. . . 4 |- -. -. x = x
7	4, 6	2false 696	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 205	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2285	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   _Vcvv 2730   \ cdif 3118   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  dfnul3  3417  rab0  3443  iotanul  5175