Theorem dfnul2 2282

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 2281	. . . 4 |- (/) = (V \ V)
2	1	eleq2i 1538	. . 3 |- (x e. (/) <-> x e. (V \ V))
3		eldif 2057	. . 3 |- (x e. (V \ V) <-> (x e. V /\ -. x e. V))
4		eqid 1475	. . . . 5 |- x = x
5		pm3.24 658	. . . . 5 |- -. (x e. V /\ -. x e. V)
6	4, 5	2th 718	. . . 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 1574	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044  (/)c0 2280

This theorem is referenced by:  dfnul3 2283  noel 2284  dm0 3323  dmsn0 3324  dmsnsn0 3325  avril1 8784

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-nul 2281

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