Theorem dfnul2 3426

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 3425	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2244	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3140	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 693	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2177	. . . . 5 |- x = x
6	5	notnoti 645	. . . 4 |- -. -. x = x
7	4, 6	2false 701	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 206	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2292	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2739   \ cdif 3128   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-nul 3425

This theorem is referenced by:  dfnul3  3427  rab0  3453  iotanul  5195