Theorem ssin

Description: Subclass of intersection. Theorem 2.8(vii) of Monk1 p. 26. (Contributed by NM, 15-Jun-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	ssin	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) ↔ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
2	1	imbi2i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) )
4		jcab	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) )
6		19.26	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) )
7	3 5 6	3bitrri	⊢ ( ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) )
8		dfss2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
9		dfss2	⊢ ( 𝐴 ⊆ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) )
10	8 9	anbi12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) ) )
11		dfss2	⊢ ( 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) )
12	7 10 11	3bitr4i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) ↔ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) )