Theorem inass

Description: Associative law for intersection of classes. Exercise 9 of TakeutiZaring p. 17. (Contributed by NM, 3-May-1994)

	Ref	Expression
Assertion	inass	|- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )

Proof

Step	Hyp	Ref	Expression
1		anass	|- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
2		elin	|- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
3	2	anbi2i	|- ( ( x e. A /\ x e. ( B i^i C ) ) <-> ( x e. A /\ ( x e. B /\ x e. C ) ) )
4	1 3	bitr4i	|- ( ( ( x e. A /\ x e. B ) /\ x e. C ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
5		elin	|- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	5	anbi1i	|- ( ( x e. ( A i^i B ) /\ x e. C ) <-> ( ( x e. A /\ x e. B ) /\ x e. C ) )
7		elin	|- ( x e. ( A i^i ( B i^i C ) ) <-> ( x e. A /\ x e. ( B i^i C ) ) )
8	4 6 7	3bitr4i	|- ( ( x e. ( A i^i B ) /\ x e. C ) <-> x e. ( A i^i ( B i^i C ) ) )
9	8	ineqri	|- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )