Theorem symdif1 3280

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 3268	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
2		difin 3252	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		incom 3207	. . . . 5 |- ( A i^i B ) = ( B i^i A )
4	3	difeq2i 3130	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
5		difin 3252	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
6	4, 5	eqtri 2115	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
7	2, 6	uneq12i 3167	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
8	1, 7	eqtr2i 2116	1 |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1296   \ cdif 3010   u. cun 3011   i^i cin 3012

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019

This theorem is referenced by: (None)