Theorem dfnul3 2286

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		pm3.24 660	. . . . 5 |- -. (x e. A /\ -. x e. A)
2		eqid 1478	. . . . 5 |- x = x
3	1, 2	2th 720	. . . 4 |- (-. (x e. A /\ -. x e. A) <-> x = x)
4	3	con1bii 220	. . 3 |- (-. x = x <-> (x e. A /\ -. x e. A))
5	4	abbii 1578	. 2 |- {x | -. x = x} = {x | (x e. A /\ -. x e. A)}
6		dfnul2 2285	. 2 |- (/) = {x | -. x = x}
7		df-rab 1655	. 2 |- {x e. A | -. x e. A} = {x | (x e. A /\ -. x e. A)}
8	5, 6, 7	3eqtr4 1508	1 |- (/) = {x e. A | -. x e. A}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  {crab 1651  (/)c0 2283

This theorem is referenced by:  kmlem3 4777

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-rab 1655  df-v 1815  df-dif 2052  df-nul 2284

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