Theorem dfnul2 3462

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 3461	. . . 4 |- (/) = ( _V \ _V )
2	1	eleq2i 2272	. . 3 |- ( x e. (/) <-> x e. ( _V \ _V ) )
3		eldif 3175	. . 3 |- ( x e. ( _V \ _V ) <-> ( x e. _V /\ -. x e. _V ) )
4		pm3.24 695	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
5		eqid 2205	. . . . 5 |- x = x
6	5	notnoti 646	. . . 4 |- -. -. x = x
7	4, 6	2false 703	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
8	2, 3, 7	3bitri 206	. 2 |- ( x e. (/) <-> -. x = x )
9	8	abbi2i 2320	1 |- (/) = { x | -. x = x }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   e. wcel 2176   {cab 2191   _Vcvv 2772   \ cdif 3163   (/)c0 3460

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-nul 3461

This theorem is referenced by:  dfnul3  3463  rab0  3489  iotanul  5247