Theorem nelbr 47978

Description: The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
nelbr	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))

Proof of Theorem nelbr

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

Step	Hyp	Ref	Expression
1		eleq12 2851	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
2	1	notbid 321	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐵))
3		df-nelbr 47976	. 2 ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
4	2, 3	brabga 5518	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   class class class wbr 5108   _∉ cnelbr 47975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-nelbr 47976

This theorem is referenced by:  nelbrim  47979  nelbrnel  47980