Theorem disj3 3475

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 3138	. . . . 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 1470	. 2 |- ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A <-> x e. ( A \ B ) ) )
6		disj1 3473	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
7		dfcleq 2171	. 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 1351   = wceq 1353   e. wcel 2148   \ cdif 3126   i^i cin 3128   (/)c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-nul 3423

This theorem is referenced by:  disjel  3477  uneqdifeqim  3508  difprsn1  3731  diftpsn3  3733  orddif  4545  phpm  6861