Theorem dfnul2 2272

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 2271	. . . 4 |- (/) = (V \ V)
2	1	eleq2i 1530	. . 3 |- (x e. (/) <-> x e. (V \ V))
3		eldif 2047	. . 3 |- (x e. (V \ V) <-> (x e. V /\ -. x e. V))
4		eqid 1468	. . . . 5 |- x = x
5		pm3.24 656	. . . . 5 |- -. (x e. V /\ -. x e. V)
6	4, 5	2th 716	. . . 4 |- (x = x <-> -. (x e. V /\ -. x e. V))
7	6	con2bii 221	. . 3 |- ((x e. V /\ -. x e. V) <-> -. x = x)
8	2, 3, 7	3bitr 177	. 2 |- (x e. (/) <-> -. x = x)
9	8	abbi2i 1566	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802   \ cdif 2034  (/)c0 2270

This theorem is referenced by:  dfnul3 2273  noel 2274  dm0 3312  dmsn0 3313  dmsnsn0 3314  avril1 8723

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-nul 2271

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