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) ∪ (B ∖ A)) = ((A ∪ B) ∖ (A ∩ B))

Proof of Theorem symdif1

Step	Hyp	Ref	Expression
1		difundir 3509	. 2 ⊢ ((A ∪ B) ∖ (A ∩ B)) = ((A ∖ (A ∩ B)) ∪ (B ∖ (A ∩ B)))
2		difin 3493	. . 3 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)
3		incom 3449	. . . . 5 ⊢ (A ∩ B) = (B ∩ A)
4	3	difeq2i 3383	. . . 4 ⊢ (B ∖ (A ∩ B)) = (B ∖ (B ∩ A))
5		difin 3493	. . . 4 ⊢ (B ∖ (B ∩ A)) = (B ∖ A)
6	4, 5	eqtri 2373	. . 3 ⊢ (B ∖ (A ∩ B)) = (B ∖ A)
7	2, 6	uneq12i 3417	. 2 ⊢ ((A ∖ (A ∩ B)) ∪ (B ∖ (A ∩ B))) = ((A ∖ B) ∪ (B ∖ A))
8	1, 7	eqtr2i 2374	1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = ((A ∪ B) ∖ (A ∩ B))

This proof depends on syntax axioms:   = wceq 1642   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This proof depends on 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 proof 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 used by: (None)