Theorem inundif 3666

Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inundif	|- ( ( A i^i B ) u. ( A \ B ) ) = A

Proof of Theorem inundif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3490	. . . 4 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2		eldif 3290	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	1, 2	orbi12i 508	. . 3 |- ( ( x e. ( A i^i B ) \/ x e. ( A \ B ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ -. x e. B ) ) )
4		pm4.42 927	. . 3 |- ( x e. A <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ -. x e. B ) ) )
5	3, 4	bitr4i 244	. 2 |- ( ( x e. ( A i^i B ) \/ x e. ( A \ B ) ) <-> x e. A )
6	5	uneqri 3449	1 |- ( ( A i^i B ) u. ( A \ B ) ) = A

Syntax hints:   -. wn 3   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   \ cdif 3277   u. cun 3278   i^i cin 3279

This theorem is referenced by:  resasplit  5572  fresaun  5573  fresaunres2  5574  ixpfi2  7363  hashun3  11613  prmreclem2  13240  sylow2a  15208  ablfac1eu  15586  basdif0  16973  neitr  17198  cmpfi  17425  ptbasfi  17566  ptcnplem  17606  fin1aufil  17917  ismbl2  19376  volinun  19393  voliunlem2  19398  mbfeqalem  19487  itg2cnlem2  19607  dvres2lem  19750  imadifxp  23991  partfun  24040  measun  24518  measunl  24523  sibfof  24607  probdif  24631  mvdco  27256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-dif 3283  df-un 3285  df-in 3287