Theorem nelbrnelim 46719

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 46717	. 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
2		df-nel 3037	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 233	1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098   ∉ wnel 3036   class class class wbr 5143   _∉ cnelbr 46713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-nelbr 46714

This theorem is referenced by: (None)