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Theorem nelbrim 43874
 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 ¬ 𝐴 ∈ 𝐵 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 43871 . . . 4 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
21relopabi 5659 . . 3 Rel _∉
32brrelex12i 5572 . 2 (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 nelbr 43873 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
54biimpd 232 . 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 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5031   _∉ cnelbr 43870 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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-nelbr 43871 This theorem is referenced by:  nelbrnelim  43876
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