Theorem untint 36570

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 4922	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		ssralv 3977	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦))
3	1, 2	syl 18	. 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦))
4	3	rexlimiv 3156	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3896  df-int 4907

This theorem is used by: (None)