Theorem disj3 2304

Description: Two ways of saying that two classes are disjoint.

Assertion

Ref	Expression
disj3	|- ((A i^i B) = (/) <-> A = (A \ B))

Proof of Theorem disj3

Step	Hyp	Ref	Expression
1		pm4.71 633	. . . 4 |- ((x e. A -> -. x e. B) <-> (x e. A <-> (x e. A /\ -. x e. B)))
2		eldif 2047	. . . . 5 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
3	2	bibi2i 606	. . . 4 |- ((x e. A <-> x e. (A \ B)) <-> (x e. A <-> (x e. A /\ -. x e. B)))
4	1, 3	bitr4 176	. . 3 |- ((x e. A -> -. x e. B) <-> (x e. A <-> x e. (A \ B)))
5	4	albii 996	. 2 |- (A.x(x e. A -> -. x e. B) <-> A.x(x e. A <-> x e. (A \ B)))
6		disj1 2302	. 2 |- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))
7		dfcleq 1463	. 2 |- (A = (A \ B) <-> A.x(x e. A <-> x e. (A \ B)))
8	5, 6, 7	3bitr4 183	1 |- ((A i^i B) = (/) <-> A = (A \ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955   \ cdif 2034   i^i cin 2036  (/)c0 2270

This theorem is referenced by:  disj4 2307  orddif 3065  php 4493  infeq5 4593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-dif 2039  df-in 2041  df-nul 2271

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