Theorem symdifcom 4220

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 4129	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
2		df-symdif 4219	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3		df-symdif 4219	. 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵))
4	1, 2, 3	3eqtr4i 2854	1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)

Syntax hints:   = wceq 1537   ∖ cdif 3933   ∪ cun 3934   △ csymdif 4218

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3941  df-symdif 4219

This theorem is referenced by:  symdifeq2  4222