Theorem dfnul2 2286

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 2285	. . . 4 |- (/) = (V \ V)
2	1	eleq2i 1541	. . 3 |- (x e. (/) <-> x e. (V \ V))
3		eldif 2061	. . 3 |- (x e. (V \ V) <-> (x e. V /\ -. x e. V))
4		eqid 1478	. . . . 5 |- x = x
5		pm3.24 660	. . . . 5 |- -. (x e. V /\ -. x e. V)
6	4, 5	2th 720	. . . 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 1577	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814   \ cdif 2048  (/)c0 2284

This theorem is referenced by:  dfnul3 2287  noel 2288  dm0 3330  dmsn0 3331  dmsnsn0 3332  avril1 8786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2053  df-nul 2285

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