Theorem symdif1 3472

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 3460	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
2		difin 3444	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		incom 3399	. . . . 5 |- ( A i^i B ) = ( B i^i A )
4	3	difeq2i 3322	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
5		difin 3444	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
6	4, 5	eqtri 2252	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
7	2, 6	uneq12i 3359	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
8	1, 7	eqtr2i 2253	1 |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1397   \ cdif 3197   u. cun 3198   i^i cin 3199

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206

This theorem is referenced by: (None)