Theorem dfsymdif2 4206

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 4205	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵))
2	1	eqabi 2895	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

This proof depends on syntax axioms:   ⊻ wxo 1541   = wceq 1570   ∈ wcel 2145  {cab 2738   △ csymdif 4197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-xor 1542  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3901  df-un 3903  df-symdif 4198

This theorem is used by: (None)