New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  symdif1 GIF version

Theorem symdif1 3519
 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 ((A B) ∪ (B A)) = ((AB) (AB))

Proof of Theorem symdif1
StepHypRef Expression
1 difundir 3508 . 2 ((AB) (AB)) = ((A (AB)) ∪ (B (AB)))
2 difin 3492 . . 3 (A (AB)) = (A B)
3 incom 3448 . . . . 5 (AB) = (BA)
43difeq2i 3382 . . . 4 (B (AB)) = (B (BA))
5 difin 3492 . . . 4 (B (BA)) = (B A)
64, 5eqtri 2373 . . 3 (B (AB)) = (B A)
72, 6uneq12i 3416 . 2 ((A (AB)) ∪ (B (AB))) = ((A B) ∪ (B A))
81, 7eqtr2i 2374 1 ((A B) ∪ (B A)) = ((AB) (AB))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator