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Theorem nelbrim 42914
Description: If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrim (𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)

Proof of Theorem nelbrim
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nelbr 42911 . . . 4 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
21relopabi 5540 . . 3 Rel _∉
32brrelex12i 5453 . 2 (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 nelbr 42913 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
54biimpd 221 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 → ¬ 𝐴𝐵))
63, 5mpcom 38 1 (𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wcel 2051  Vcvv 3408   class class class wbr 4925   _∉ cnelbr 42910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-nelbr 42911
This theorem is referenced by:  nelbrnelim  42916
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