Theorem symdif1 3392

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 ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Proof of Theorem symdif1

Step	Hyp	Ref	Expression
1		difundir 3380	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
2		difin 3364	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		incom 3319	. . . . 5 |- ( A i^i B ) = ( B i^i A )
4	3	difeq2i 3242	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
5		difin 3364	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
6	4, 5	eqtri 2191	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
7	2, 6	uneq12i 3279	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
8	1, 7	eqtr2i 2192	1 |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1348   \ cdif 3118   u. cun 3119   i^i cin 3120

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127

This theorem is referenced by: (None)