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Theorem dfsymdif3 4266
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 4235 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
2 incom 4175 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
32difeq2i 4093 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
4 difin 4235 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
53, 4eqtri 2841 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
61, 5uneq12i 4134 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
7 difundir 4254 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
8 df-symdif 4216 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
96, 7, 83eqtr4ri 2852 1 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cdif 3930  cun 3931  cin 3932  csymdif 4215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-symdif 4216
This theorem is referenced by: (None)
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