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Metamath Proof Explorer

Theorem dfsymdif3

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)

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

Proof

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