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Theorem symdif1 3373
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 3361 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
2 difin 3345 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 incom 3300 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43difeq2i 3223 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
5 difin 3345 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
64, 5eqtri 2178 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
72, 6uneq12i 3260 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
81, 7eqtr2i 2179 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:   = wceq 1335  cdif 3099  cun 3100  cin 3101
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108
This theorem is referenced by: (None)
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