Theorem dfnul2 3917

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 3916	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2693	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3584	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		eqid 2622	. . . . 5 |- x = x
5		pm3.24 926	. . . . 5 |- -. ( x e. _V /\ -. x e. _V )
6	4, 5	2th 254	. . . 4 |- ( x = x <-> -. ( x e. _V /\ -. x e. _V ) )
7	6	con2bii 347	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 286	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2738	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  dfnul3  3918  rab0OLD  3956  iotanul  5866  avril1  27319