Theorem nelbrnelim 42992

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 42990	. 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
2		df-nel 3091	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 235	1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2081   ∉ wnel 3090   class class class wbr 4962   _∉ cnelbr 42986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-nelbr 42987

This theorem is referenced by: (None)