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 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) )
2		elin	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
3	2	anbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) )
4	1 3	bitr4i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) )
5		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
6	5	anbi1i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∧ 𝑥 ∈ 𝐶 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐶 ) )
7		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) )
8	4 6 7	3bitr4i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∧ 𝑥 ∈ 𝐶 ) ↔ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) )
9	8	ineqri	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) )