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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelbrnelim | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nelbrnelim | ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelbrim 46560 | . 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) | |
2 | df-nel 3041 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2098 ∉ wnel 3040 class class class wbr 5141 _∉ cnelbr 46556 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-nelbr 46557 |
This theorem is referenced by: (None) |
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