Theorem inundif 3629

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 3220	. . . 4 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
2		eldif 3222	. . . 4 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
3	1, 2	orbi12i 507	. . 3 |- ((x e. (A i^i B) \/ x e. (A \ B)) <-> ((x e. A /\ x e. B) \/ (x e. A /\ -. x e. B)))
4		pm4.42 926	. . 3 |- (x e. A <-> ((x e. A /\ x e. B) \/ (x e. A /\ -. x e. B)))
5	3, 4	bitr4i 243	. 2 |- ((x e. (A i^i B) \/ x e. (A \ B)) <-> x e. A)
6	5	uneqri 3407	1 |- ((A i^i B) u. (A \ B)) = A

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3207   u. cun 3208   i^i cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is referenced by:  phialllem2  4618  sbthlem1  6204