Theorem dfsymdif3 3893

Description: Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.)

Assertion

Ref	Expression
dfsymdif3	|- ( A /_\ B ) = ( ( A u. B ) \ ( A i^i B ) )

Proof of Theorem dfsymdif3

Step	Hyp	Ref	Expression
1		difin 3861	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
2		incom 3805	. . . . 5 |- ( A i^i B ) = ( B i^i A )
3	2	difeq2i 3725	. . . 4 |- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
4		difin 3861	. . . 4 |- ( B \ ( B i^i A ) ) = ( B \ A )
5	3, 4	eqtri 2644	. . 3 |- ( B \ ( A i^i B ) ) = ( B \ A )
6	1, 5	uneq12i 3765	. 2 |- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
7		difundir 3880	. 2 |- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
8		df-symdif 3844	. 2 |- ( A /_\ B ) = ( ( A \ B ) u. ( B \ A ) )
9	6, 7, 8	3eqtr4ri 2655	1 |- ( A /_\ B ) = ( ( A u. B ) \ ( A i^i B ) )

Syntax hints:   = wceq 1483   \ cdif 3571   u. cun 3572   i^i cin 3573   /_\ csymdif 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-symdif 3844

This theorem is referenced by: (None)