Theorem untint

Description: If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011)

	Ref	Expression
Assertion	untint	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		intss1	⊢ ( 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥 )
2		ssralv	⊢ ( ∩ 𝐴 ⊆ 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) )
3	1 2	syl	⊢ ( 𝑥 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) )
4	3	rexlimiv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 )