Theorem symdif1 1689

Description: Two ways of expressing 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) ∪ (B ∖ A)) = ((A ∪ B) ∖ (A ∩ B))

Proof of Theorem symdif1

Step	Hyp	Ref	Expression
1		difundir 1682	. 2 ⊢ ((A ∪ B) ∖ (A ∩ B)) = ((A ∖ (A ∩ B)) ∪ (B ∖ (A ∩ B)))
2		difin 1670	. . 3 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)
3		incom 1636	. . . . 5 ⊢ (A ∩ B) = (B ∩ A)
4	3	difeq2i 1585	. . . 4 ⊢ (B ∖ (A ∩ B)) = (B ∖ (B ∩ A))
5		difin 1670	. . . 4 ⊢ (B ∖ (B ∩ A)) = (B ∖ A)
6	4, 5	eqtr 1119	. . 3 ⊢ (B ∖ (A ∩ B)) = (B ∖ A)
7	2, 6	uneq12i 1609	. 2 ⊢ ((A ∖ (A ∩ B)) ∪ (B ∖ (A ∩ B))) = ((A ∖ B) ∪ (B ∖ A))
8	1, 7	eqtr2 1120	1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = ((A ∪ B) ∖ (A ∩ B))

Syntax hints:   = wceq 1091   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491

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