Theorem symdif1 3437

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 3425	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
2		difin 3409	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		incom 3364	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	difeq2i 3287	. . . 4 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ (𝐵 ∩ 𝐴))
5		difin 3409	. . . 4 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
6	4, 5	eqtri 2225	. . 3 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
7	2, 6	uneq12i 3324	. 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
8	1, 7	eqtr2i 2226	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))

Syntax hints:   = wceq 1372   ∖ cdif 3162   ∪ cun 3163   ∩ cin 3164

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171

This theorem is referenced by: (None)