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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 )