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Metamath Proof Explorer

Theorem ssiin

Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003)

Ref Expression
Assertion ssiin ⊢ C ⊆ ⋂ x ∈ A B ↔ ∀ x ∈ A C ⊆ B

Proof

Step Hyp Ref Expression
1 nfcv ⊢ Ⅎ _ x C
2 1 ssiinf ⊢ C ⊆ ⋂ x ∈ A B ↔ ∀ x ∈ A C ⊆ B