Theorem disj3 3499

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		pm4.71 389	. . . 4 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
2		eldif 3162	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	2	bibi2i 227	. . . 4 |- ( ( x e. A <-> x e. ( A \ B ) ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
4	1, 3	bitr4i 187	. . 3 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> x e. ( A \ B ) ) )
5	4	albii 1481	. 2 |- ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
6		disj1 3497	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
7		dfcleq 2187	. 2 |- ( A = ( A \ B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
8	5, 6, 7	3bitr4i 212	1 |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   \ cdif 3150   i^i cin 3152   (/)c0 3446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-in 3159  df-nul 3447

This theorem is referenced by:  disjel  3501  uneqdifeqim  3532  difprsn1  3757  diftpsn3  3759  orddif  4579  phpm  6921