Theorem symdif1 3520

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 3509	. 2 |- ((A u. B) \ (A i^i B)) = ((A \ (A i^i B)) u. (B \ (A i^i B)))
2		difin 3493	. . 3 |- (A \ (A i^i B)) = (A \ B)
3		incom 3449	. . . . 5 |- (A i^i B) = (B i^i A)
4	3	difeq2i 3383	. . . 4 |- (B \ (A i^i B)) = (B \ (B i^i A))
5		difin 3493	. . . 4 |- (B \ (B i^i A)) = (B \ A)
6	4, 5	eqtri 2373	. . 3 |- (B \ (A i^i B)) = (B \ A)
7	2, 6	uneq12i 3417	. 2 |- ((A \ (A i^i B)) u. (B \ (A i^i B))) = ((A \ B) u. (B \ A))
8	1, 7	eqtr2i 2374	1 |- ((A \ B) u. (B \ A)) = ((A u. B) \ (A i^i B))

Syntax hints:   = wceq 1642   \ 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: (None)