Theorem untint 33653

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.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untint

Step	Hyp	Ref	Expression
1		intss1 4894	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		ssralv 3987	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦))
3	1, 2	syl 17	. 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦))
4	3	rexlimiv 3209	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∩ cint 4879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-int 4880

This theorem is referenced by: (None)