Theorem symdifcom 3543

Description: Symmetric difference commutes. (Contributed by SF, 11-Jan-2015.)

Assertion

Ref	Expression
symdifcom	⊢ (A ⊕ B) = (B ⊕ A)

Proof of Theorem symdifcom

Step	Hyp	Ref	Expression
1		uncom 3409	. 2 ⊢ ((A ∖ B) ∪ (B ∖ A)) = ((B ∖ A) ∪ (A ∖ B))
2		df-symdif 3217	. 2 ⊢ (A ⊕ B) = ((A ∖ B) ∪ (B ∖ A))
3		df-symdif 3217	. 2 ⊢ (B ⊕ A) = ((B ∖ A) ∪ (A ∖ B))
4	1, 2, 3	3eqtr4i 2383	1 ⊢ (A ⊕ B) = (B ⊕ A)

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∪ cun 3208   ⊕ csymdif 3210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-symdif 3217

This theorem is referenced by: (None)