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Theorem dfsymdif3 4036
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 4004 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
2 incom 3948 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
32difeq2i 3868 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
4 difin 4004 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
53, 4eqtri 2782 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
61, 5uneq12i 3908 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
7 difundir 4023 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
8 df-symdif 3987 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
96, 7, 83eqtr4ri 2793 1 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cdif 3712  cun 3713  cin 3714  csymdif 3986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-symdif 3987
This theorem is referenced by: (None)
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