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Theorem symdif1 3311
Description: Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
symdif1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))

Proof of Theorem symdif1
StepHypRef Expression
1 difundir 3299 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
2 difin 3283 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 incom 3238 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43difeq2i 3161 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
5 difin 3283 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
64, 5eqtri 2138 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
72, 6uneq12i 3198 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
81, 7eqtr2i 2139 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:   = wceq 1316  cdif 3038  cun 3039  cin 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047
This theorem is referenced by: (None)
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