Theorem inundifOLD 2952

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.)

Assertion

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

Proof of Theorem inundifOLD

Step	Hyp	Ref	Expression
1		undir 2843	. 2 |- ((A i^i B) u. (A \ B)) = ((A u. (A \ B)) i^i (B u. (A \ B)))
2		difss 2735	. . . 4 |- (A \ B) C_ A
3		ssequn2 2779	. . . 4 |- ((A \ B) C_ A <-> (A u. (A \ B)) = A)
4	2, 3	mpbi 206	. . 3 |- (A u. (A \ B)) = A
5		undif2 2950	. . . 4 |- (B u. (A \ B)) = (B u. A)
6		uncom 2744	. . . 4 |- (B u. A) = (A u. B)
7	5, 6	eqtri 1908	. . 3 |- (B u. (A \ B)) = (A u. B)
8	4, 7	ineq12i 2794	. 2 |- ((A u. (A \ B)) i^i (B u. (A \ B))) = (A i^i (A u. B))
9		inabs 2823	. 2 |- (A i^i (A u. B)) = A
10	1, 8, 9	3eqtri 1912	1 |- ((A i^i B) u. (A \ B)) = A

Syntax hints:   = wceq 1298   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876

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