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Metamath Proof Explorer

Theorem dfsymdif2

Description: Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020)

Ref Expression
Assertion dfsymdif2 ( 𝐴 △ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵 ) }

Proof

Step Hyp Ref Expression
1 elsymdifxor ⊢ ( 𝑥 ∈ ( 𝐴 △ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵 ) )
2 1 eqabi ⊢ ( 𝐴 △ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵 ) }