Theorem symdifid 4977

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 4149	. 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴))
2		difid 4271	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
3	2, 2	uneq12i 4068	. 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅)
4		un0 4289	. 2 ⊢ (∅ ∪ ∅) = ∅
5	1, 3, 4	3eqtri 2785	1 ⊢ (𝐴 △ 𝐴) = ∅

Syntax hints:   = wceq 1538   ∖ cdif 3857   ∪ cun 3858   △ csymdif 4148  ∅c0 4227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-symdif 4149  df-nul 4228

This theorem is referenced by: (None)