Theorem symdifcom 4199

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 4103	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
2		df-symdif 4198	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3		df-symdif 4198	. 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
4	1, 2, 3	3eqtr4i 2764	1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)

Syntax hints:   = wceq 1541   ∖ cdif 3894   ∪ cun 3895   △ csymdif 4197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-symdif 4198

This theorem is referenced by:  symdifeq2  4201