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Theorem untint 35712
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 4963 . . 3 (𝑥𝐴 𝐴𝑥)
2 ssralv 4052 . . 3 ( 𝐴𝑥 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
31, 2syl 17 . 2 (𝑥𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦))
43rexlimiv 3148 1 (∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2108  wral 3061  wrex 3070  wss 3951   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-int 4947
This theorem is referenced by: (None)
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