Theorem difundi

Description: Distributive law for class difference. Theorem 39 of Suppes p. 29. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	difundi	⊢ ( 𝐴 ∖ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		dfun3	⊢ ( 𝐵 ∪ 𝐶 ) = ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) )
2	1	difeq2i	⊢ ( 𝐴 ∖ ( 𝐵 ∪ 𝐶 ) ) = ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) )
3		inindi	⊢ ( 𝐴 ∩ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴 ∩ ( V ∖ 𝐵 ) ) ∩ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) )
4		dfin2	⊢ ( 𝐴 ∩ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) = ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) )
5		invdif	⊢ ( 𝐴 ∩ ( V ∖ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )
6		invdif	⊢ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) = ( 𝐴 ∖ 𝐶 )
7	5 6	ineq12i	⊢ ( ( 𝐴 ∩ ( V ∖ 𝐵 ) ) ∩ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∖ 𝐶 ) )
8	3 4 7	3eqtr3i	⊢ ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∖ 𝐶 ) )
9	2 8	eqtri	⊢ ( 𝐴 ∖ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∖ 𝐶 ) )