Theorem uncom

Description: Commutative law for union of classes. Exercise 6 of TakeutiZaring p. 17. (Contributed by NM, 25-Jun-1998) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	uncom	|- ( A u. B ) = ( B u. A )

Proof

Step	Hyp	Ref	Expression
1		orcom	|- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun	|- ( x e. ( B u. A ) <-> ( x e. B \/ x e. A ) )
3	1 2	bitr4i	|- ( ( x e. A \/ x e. B ) <-> x e. ( B u. A ) )
4	3	uneqri	|- ( A u. B ) = ( B u. A )