Theorem untint 31589

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 4492	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		ssralv 3666	. . 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 3027	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by: (None)