Theorem ssint

Description: Subclass of a class intersection. Theorem 5.11(viii) of Monk1 p. 52 and its converse. (Contributed by NM, 14-Oct-1999)

	Ref	Expression
Assertion	ssint	⊢ ( 𝐴 ⊆ ∩ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		dfss3	⊢ ( 𝐴 ⊆ ∩ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 )
2		vex	⊢ 𝑦 ∈ V
3	2	elint2	⊢ ( 𝑦 ∈ ∩ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 )
4	3	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 )
5		ralcom	⊢ ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝑥 )
6		dfss3	⊢ ( 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝑥 )
7	6	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝑥 )
8	5 7	bitr4i	⊢ ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 )
9	1 4 8	3bitri	⊢ ( 𝐴 ⊆ ∩ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 )