ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  symdif1 GIF version

Theorem symdif1 3264
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 3252 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
2 difin 3236 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 incom 3192 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43difeq2i 3115 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
5 difin 3236 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
64, 5eqtri 2108 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
72, 6uneq12i 3152 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
81, 7eqtr2i 2109 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cdif 2996  cun 2997  cin 2998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator