Theorem symdifid 5054

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 4214	. 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴))
2		difid 4338	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
3	2, 2	uneq12i 4128	. 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅)
4		un0 4357	. 2 ⊢ (∅ ∪ ∅) = ∅
5	1, 3, 4	3eqtri 2796	1 ⊢ (𝐴 △ 𝐴) = ∅

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∪ cun 3911   △ csymdif 4213  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-symdif 4214  df-nul 4295

This theorem is referenced by: (None)