Theorem symdif1 2265

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.

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 2258	. 2 |- ((A u. B) \ (A i^i B)) = ((A \ (A i^i B)) u. (B \ (A i^i B)))
2		difin 2245	. . 3 |- (A \ (A i^i B)) = (A \ B)
3		incom 2208	. . . . 5 |- (A i^i B) = (B i^i A)
4	3	difeq2i 2156	. . . 4 |- (B \ (A i^i B)) = (B \ (B i^i A))
5		difin 2245	. . . 4 |- (B \ (B i^i A)) = (B \ A)
6	4, 5	eqtr 1495	. . 3 |- (B \ (A i^i B)) = (B \ A)
7	2, 6	uneq12i 2182	. 2 |- ((A \ (A i^i B)) u. (B \ (A i^i B))) = ((A \ B) u. (B \ A))
8	1, 7	eqtr2 1496	1 |- ((A \ B) u. (B \ A)) = ((A u. B) \ (A i^i B))

Syntax hints:   = wceq 956   \ cdif 2044   u. cun 2045   i^i cin 2046

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-in 2051

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