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Theorem nelbrnelim 44441
Description: If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrnelim (𝐴 _∉ 𝐵𝐴𝐵)

Proof of Theorem nelbrnelim
StepHypRef Expression
1 nelbrim 44439 . 2 (𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)
2 df-nel 3047 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
31, 2sylibr 237 1 (𝐴 _∉ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2110  wnel 3046   class class class wbr 5053   _∉ cnelbr 44435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-nelbr 44436
This theorem is referenced by: (None)
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