Theorem symdifid 5044

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 4207	. 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴))
2		difid 4330	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
3	2, 2	uneq12i 4120	. 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅)
4		un0 4348	. 2 ⊢ (∅ ∪ ∅) = ∅
5	1, 3, 4	3eqtri 2764	1 ⊢ (𝐴 △ 𝐴) = ∅

Syntax hints:   = wceq 1542   ∖ cdif 3900   ∪ cun 3901   △ csymdif 4206  ∅c0 4287

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-symdif 4207  df-nul 4288

This theorem is referenced by: (None)