Theorem dfnul3 2335

Description: Alternate definition of the empty set..

Assertion

Ref	Expression
dfnul3	|- (/) = {x e. A | -. x e. A}

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		pm3.24 661	. . . . 5 |- -. (x e. A /\ -. x e. A)
2		eqid 1518	. . . . 5 |- x = x
3	1, 2	2th 723	. . . 4 |- (-. (x e. A /\ -. x e. A) <-> x = x)
4	3	con1bii 218	. . 3 |- (-. x = x <-> (x e. A /\ -. x e. A))
5	4	abbii 1618	. 2 |- {x | -. x = x} = {x | (x e. A /\ -. x e. A)}
6		dfnul2 2334	. 2 |- (/) = {x | -. x = x}
7		df-rab 1698	. 2 |- {x e. A | -. x e. A} = {x | (x e. A /\ -. x e. A)}
8	5, 6, 7	3eqtr4i 1548	1 |- (/) = {x e. A | -. x e. A}

Syntax hints:  -. wn 2   /\ wa 221   = wceq 992   e. wcel 994  {cab 1505  {crab 1694  (/)c0 2332

This theorem is referenced by:  kmlem3 4913

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-rab 1698  df-v 1858  df-dif 2101  df-nul 2333

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