Theorem dfsymdif2 4213

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfsymdif2

Step	Hyp	Ref	Expression
1		elsymdifxor 4212	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵))
2	1	eqabi 2896	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Syntax hints:   ⊻ wxo 1539   = wceq 1568   ∈ wcel 2141  {cab 2739   △ csymdif 4204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-xor 1540  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-un 3909  df-symdif 4205

This theorem is referenced by: (None)