Theorem dfun3

Description: Union defined in terms of intersection (De Morgan's law). Definition of union in Mendelson p. 231. (Contributed by NM, 8-Jan-2002)

	Ref	Expression
Assertion	dfun3	|- ( A u. B ) = ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfun2	|- ( A u. B ) = ( _V \ ( ( _V \ A ) \ B ) )
2		dfin2	|- ( ( _V \ A ) i^i ( _V \ B ) ) = ( ( _V \ A ) \ ( _V \ ( _V \ B ) ) )
3		ddif	|- ( _V \ ( _V \ B ) ) = B
4	3	difeq2i	|- ( ( _V \ A ) \ ( _V \ ( _V \ B ) ) ) = ( ( _V \ A ) \ B )
5	2 4	eqtr2i	|- ( ( _V \ A ) \ B ) = ( ( _V \ A ) i^i ( _V \ B ) )
6	5	difeq2i	|- ( _V \ ( ( _V \ A ) \ B ) ) = ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) )
7	1 6	eqtri	|- ( A u. B ) = ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) )