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Metamath Proof Explorer

Theorem symdif0

Description: Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012)

Ref Expression
Assertion symdif0 ⊢ A ∆ ∅ = A

Proof

Step Hyp Ref Expression
1 df-symdif ⊢ A ∆ ∅ = A ∖ ∅ ∪ ∅ ∖ A
2 dif0 ⊢ A ∖ ∅ = A
3 0dif ⊢ ∅ ∖ A = ∅
4 2 3 uneq12i ⊢ A ∖ ∅ ∪ ∅ ∖ A = A ∪ ∅
5 un0 ⊢ A ∪ ∅ = A
6 1 4 5 3eqtri ⊢ A ∆ ∅ = A