Theorem indif

Description: Intersection with class difference. Theorem 34 of Suppes p. 29. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	indif	⊢ ( 𝐴 ∩ ( 𝐴 ∖ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dfin4	⊢ ( 𝐴 ∩ ( 𝐴 ∖ 𝐵 ) ) = ( 𝐴 ∖ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) )
2		dfin4	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) )
3	2	difeq2i	⊢ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) )
4		difin	⊢ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )
5	1 3 4	3eqtr2i	⊢ ( 𝐴 ∩ ( 𝐴 ∖ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )