Theorem nelbrim 47252

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 47249	. . . 4 ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
2	1	relopabiv 5799	. . 3 ⊢ Rel _∉
3	2	brrelex12i 5709	. 2 ⊢ (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		nelbr 47251	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))
5	4	biimpd 229	. 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 3459   class class class wbr 5119   _∉ cnelbr 47248

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-nelbr 47249

This theorem is referenced by:  nelbrnelim  47254