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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 ∪ 𝐵 ) ∖ ( 𝐴 ∩ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 difin ⊢ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )
2 incom ⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
3 2 difeq2i ⊢ ( 𝐵 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐵 ∖ ( 𝐵 ∩ 𝐴 ) )
4 difin ⊢ ( 𝐵 ∖ ( 𝐵 ∩ 𝐴 ) ) = ( 𝐵 ∖ 𝐴 )
5 3 4 eqtri ⊢ ( 𝐵 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐵 ∖ 𝐴 )
6 1 5 uneq12i ⊢ ( ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) ∪ ( 𝐵 ∖ ( 𝐴 ∩ 𝐵 ) ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐴 ) )
7 difundir ⊢ ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) ∪ ( 𝐵 ∖ ( 𝐴 ∩ 𝐵 ) ) )
8 df-symdif ⊢ ( 𝐴 △ 𝐵 ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐴 ) )
9 6 7 8 3eqtr4ri ⊢ ( 𝐴 △ 𝐵 ) = ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐴 ∩ 𝐵 ) )