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Theorem dfsymdif3 4305
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 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))

Proof of Theorem dfsymdif3
StepHypRef Expression
1 difin 4271 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
2 incom 4208 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
32difeq2i 4122 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
4 difin 4271 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
53, 4eqtri 2764 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
61, 5uneq12i 4165 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
7 difundir 4290 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
8 df-symdif 4252 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
96, 7, 83eqtr4ri 2775 1 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3947  cun 3948  cin 3949  csymdif 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-symdif 4252
This theorem is referenced by: (None)
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