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Theorem dfsymdif3 3852
 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 3823 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
2 incom 3767 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
32difeq2i 3687 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
4 difin 3823 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
53, 4eqtri 2632 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
61, 5uneq12i 3727 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
7 difundir 3839 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
8 df-symdif 3806 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
96, 7, 83eqtr4ri 2643 1 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   △ csymdif 3805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-symdif 3806 This theorem is referenced by: (None)
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