Theorem dfnul2 2332

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 2331	. . . 4 |- (/) = (V \ V)
2	1	eleq2i 1579	. . 3 |- (x e. (/) <-> x e. (V \ V))
3		eldif 2107	. . 3 |- (x e. (V \ V) <-> (x e. V /\ -. x e. V))
4		eqid 1516	. . . . 5 |- x = x
5		pm3.24 660	. . . . 5 |- -. (x e. V /\ -. x e. V)
6	4, 5	2th 721	. . . 4 |- (x = x <-> -. (x e. V /\ -. x e. V))
7	6	con2bii 219	. . 3 |- ((x e. V /\ -. x e. V) <-> -. x = x)
8	2, 3, 7	3bitri 175	. 2 |- (x e. (/) <-> -. x = x)
9	8	abbi2i 1615	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 2   /\ wa 221   = wceq 989   e. wcel 991  {cab 1503  Vcvv 1855   \ cdif 2094  (/)c0 2330

This theorem is referenced by:  dfnul3 2333  noel 2334  dm0 3411  avril1 9035

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 995  ax-gen 996  ax-8 997  ax-10 999  ax-12 1001  ax-17 1004  ax-4 1006  ax-5o 1008  ax-6o 1011  ax-9o 1156  ax-10o 1174  ax-16 1244  ax-11o 1252  ax-ext 1498

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1014  df-sb 1206  df-clab 1504  df-cleq 1509  df-clel 1512  df-v 1856  df-dif 2099  df-nul 2331

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