Theorem nelbrim 44654

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 → ¬ 𝐴 ∈ 𝐵)

Proof of Theorem nelbrim

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nelbr 44651	. . . 4 ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
2	1	relopabiv 5719	. . 3 ⊢ Rel _∉
3	2	brrelex12i 5633	. 2 ⊢ (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		nelbr 44653	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))
5	4	biimpd 228	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵))
6	3, 5	mpcom 38	1 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070   _∉ cnelbr 44650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-nelbr 44651

This theorem is referenced by:  nelbrnelim  44656