Theorem symdif1 3387

Description: Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
symdif1	|- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Proof of Theorem symdif1

Step	Hyp	Ref	Expression
1		difundir 3375	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
2		difin 3359	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		incom 3314	. . . . 5 |- ( A i^i B ) = ( B i^i A )
4	3	difeq2i 3237	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
5		difin 3359	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
6	4, 5	eqtri 2186	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
7	2, 6	uneq12i 3274	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
8	1, 7	eqtr2i 2187	1 |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1343   \ cdif 3113   u. cun 3114   i^i cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122

This theorem is referenced by: (None)