Theorem dfsymdif3 4253

Description: Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.)

Assertion

Ref	Expression
dfsymdif3	⊢ (𝐴 △ 𝐵) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))

Proof of Theorem dfsymdif3

Step	Hyp	Ref	Expression
1		difin 4219	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
2		incom 4156	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	difeq2i 4070	. . . 4 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ (𝐵 ∩ 𝐴))
4		difin 4219	. . . 4 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
5	3, 4	eqtri 2754	. . 3 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
6	1, 5	uneq12i 4113	. 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
7		difundir 4238	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
8		df-symdif 4200	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
9	6, 7, 8	3eqtr4ri 2765	1 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))

Syntax hints:   = wceq 1541   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   △ csymdif 4199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-symdif 4200

This theorem is referenced by: (None)