Theorem symdifcom 4234

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 4138	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
2		df-symdif 4233	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3		df-symdif 4233	. 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
4	1, 2, 3	3eqtr4i 2769	1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)

Syntax hints:   = wceq 1540   ∖ cdif 3928   ∪ cun 3929   △ csymdif 4232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-symdif 4233

This theorem is referenced by:  symdifeq2  4236