Theorem nelbrnelim 48550

Description: If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
nelbrnelim	⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

Proof of Theorem nelbrnelim

Step	Hyp	Ref	Expression
1		nelbrim 48548	. 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
2		df-nel 3066	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 237	1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ∉ wnel 3065   class class class wbr 5106   _∉ cnelbr 48544

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-nelbr 48545

This theorem is used by: (None)