Theorem nelbrim 48267

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 48264	. . . 4 ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
2	1	relopabiv 5794	. . 3 ⊢ Rel _∉
3	2	brrelex12i 5702	. 2 ⊢ (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		nelbr 48266	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))
5	4	biimpd 232	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵))
6	3, 5	mpcom 39	1 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145  Vcvv 3450   class class class wbr 5102   _∉ cnelbr 48263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-nelbr 48264

This theorem is used by:  nelbrnelim  48269