Theorem disj3 3272

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		pm4.71 369	. . . 4 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
2		eldif 2927	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	2	bibi2i 216	. . . 4 |- ( ( x e. A <-> x e. ( A \ B ) ) <-> ( x e. A <-> ( x e. A /\ -. x e. B ) ) )
4	1, 3	bitr4i 176	. . 3 |- ( ( x e. A -> -. x e. B ) <-> ( x e. A <-> x e. ( A \ B ) ) )
5	4	albii 1359	. 2 |- ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
6		disj1 3270	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
7		dfcleq 2034	. 2 |- ( A = ( A \ B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
8	5, 6, 7	3bitr4i 201	1 |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   \ cdif 2914   i^i cin 2916   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225

This theorem is referenced by:  disjel  3274  disj4im  3276  uneqdifeqim  3308  difprsn1  3503  diftpsn3  3505  orddif  4271  phpm  6327