Theorem nelbrnelim 43483

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 43481	. 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
2		df-nel 3126	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 236	1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ∉ wnel 3125   class class class wbr 5068   _∉ cnelbr 43477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-nelbr 43478

This theorem is referenced by: (None)