Theorem dfnul2 2877

Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20.

Assertion

Ref	Expression
dfnul2	|- (/) = {x | -. x = x}

Proof of Theorem dfnul2

Step	Hyp	Ref	Expression
1		df-nul 2876	. . . 4 |- (/) = (_V \ _V)
2	1	eleq2i 1961	. . 3 |- (x e. (/) <-> x e. (_V \ _V))
3		eldif 2609	. . 3 |- (x e. (_V \ _V) <-> (x e. _V /\ -. x e. _V))
4		eqid 1884	. . . . 5 |- x = x
5		pm3.24 720	. . . . 5 |- -. (x e. _V /\ -. x e. _V)
6	4, 5	2th 786	. . . 4 |- (x = x <-> -. (x e. _V /\ -. x e. _V))
7	6	con2bii 238	. . 3 |- ((x e. _V /\ -. x e. _V) <-> -. x = x)
8	2, 3, 7	3bitri 194	. 2 |- (x e. (/) <-> -. x = x)
9	8	abbi2i 2005	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590  (/)c0 2875

This theorem is referenced by:  dfnul3 2878  noel 2879  rab0 2894  dm0OLD 4171  iotanul 5098  avril1 10142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-nul 2876

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