Theorem dfnul2 3439

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 3438	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2256	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3153	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 694	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2189	. . . . 5 |- x = x
6	5	notnoti 646	. . . 4 |- -. -. x = x
7	4, 6	2false 702	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 206	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2304	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   e. wcel 2160   {cab 2175   _Vcvv 2752   \ cdif 3141   (/)c0 3437

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438

This theorem is referenced by:  dfnul3  3440  rab0  3466  iotanul  5211