Theorem dfnul2 3498

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 3497	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2298	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3210	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 701	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2231	. . . . 5 |- x = x
6	5	notnoti 650	. . . 4 |- -. -. x = x
7	4, 6	2false 709	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 206	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2346	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   _Vcvv 2803   \ cdif 3198   (/)c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-nul 3497

This theorem is referenced by:  dfnul3  3499  rab0  3525  iotanul  5309