Theorem unvdif

Description: The union of a class and its complement is the universe. Theorem 5.1(5) of Stoll p. 17. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	unvdif	|- ( A u. ( _V \ A ) ) = _V

Proof

Step	Hyp	Ref	Expression
1		dfun3	|- ( A u. ( _V \ A ) ) = ( _V \ ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) )
2		disjdif	|- ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) = (/)
3	2	difeq2i	|- ( _V \ ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) ) = ( _V \ (/) )
4		dif0	|- ( _V \ (/) ) = _V
5	1 3 4	3eqtri	|- ( A u. ( _V \ A ) ) = _V