Theorem symdif1 3202

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 3190	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
2		difin 3174	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		incom 3129	. . . . 5 |- ( A i^i B ) = ( B i^i A )
4	3	difeq2i 3059	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
5		difin 3174	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
6	4, 5	eqtri 2060	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
7	2, 6	uneq12i 3095	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
8	1, 7	eqtr2i 2061	1 |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1243   \ cdif 2914   u. cun 2915   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924

This theorem is referenced by: (None)