Theorem symdifcom 4204

Description: Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)

Assertion

Ref	Expression
symdifcom	⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)

Proof of Theorem symdifcom

Step	Hyp	Ref	Expression
1		uncom 4109	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
2		df-symdif 4203	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3		df-symdif 4203	. 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
4	1, 2, 3	3eqtr4i 2794	1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)

Syntax hints:   = wceq 1559   ∖ cdif 3899   ∪ cun 3900   △ csymdif 4202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-symdif 4203

This theorem is referenced by:  symdifeq2  4206