Theorem symdifid 4599

Description: Symmetric difference with self yields 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 3844	. 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴))
2		difid 3948	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
3	2, 2	uneq12i 3765	. 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅)
4		un0 3967	. 2 ⊢ (∅ ∪ ∅) = ∅
5	1, 3, 4	3eqtri 2648	1 ⊢ (𝐴 △ 𝐴) = ∅

Syntax hints:   = wceq 1483   ∖ cdif 3571   ∪ cun 3572   △ csymdif 3843  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916

This theorem is referenced by: (None)