Theorem nelbrnelim 46651

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 46649	. 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
2		df-nel 3043	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 233	1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2099   ∉ wnel 3042   class class class wbr 5142   _∉ cnelbr 46645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-nelbr 46646

This theorem is referenced by: (None)