Theorem disj3 3410

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Assertion

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

Proof of Theorem disj3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm4.71 386	. . . 4 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
2		eldif 3075	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	2	bibi2i 226	. . . 4 |- ( ( x e. A <-> x e. ( A \ B ) ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
4	1, 3	bitr4i 186	. . 3 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> x e. ( A \ B ) ) )
5	4	albii 1446	. 2 |- ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
6		disj1 3408	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
7		dfcleq 2131	. 2 |- ( A = ( A \ B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
8	5, 6, 7	3bitr4i 211	1 |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   \ cdif 3063   i^i cin 3065   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359

This theorem is referenced by:  disjel  3412  uneqdifeqim  3443  difprsn1  3654  diftpsn3  3656  orddif  4457  phpm  6752