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Theorem untint 33169
 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 4853 . . 3 (𝑥𝐴 𝐴𝑥)
2 ssralv 3958 . . 3 ( 𝐴𝑥 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
31, 2syl 17 . 2 (𝑥𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
43rexlimiv 3204 1 (∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3858  ∩ cint 4838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-in 3865  df-ss 3875  df-int 4839 This theorem is referenced by: (None)
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