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Theorem untint 30843
 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 (∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untint
StepHypRef Expression
1 intss1 4427 . . 3 (𝑥𝐴 𝐴𝑥)
2 ssralv 3629 . . 3 ( 𝐴𝑥 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
31, 2syl 17 . 2 (𝑥𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
43rexlimiv 3009 1 (∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by: (None)
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