Theorem symdifid 5067

Description: The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)

Assertion

Ref	Expression
symdifid	⊢ (𝐴 △ 𝐴) = ∅

Proof of Theorem symdifid

Step	Hyp	Ref	Expression
1		df-symdif 4233	. 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴))
2		difid 4356	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
3	2, 2	uneq12i 4146	. 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅)
4		un0 4374	. 2 ⊢ (∅ ∪ ∅) = ∅
5	1, 3, 4	3eqtri 2761	1 ⊢ (𝐴 △ 𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3928   ∪ cun 3929   △ csymdif 4232  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-symdif 4233  df-nul 4314

This theorem is referenced by: (None)