Theorem nelbr 47866

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 2855	. . 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 47864	. 2 ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
4	2, 3	brabga 5509	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   class class class wbr 5105   _∉ cnelbr 47863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-nelbr 47864

This theorem is referenced by:  nelbrim  47867  nelbrnel  47868