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Mathbox for Alexander van der Vekens |
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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 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) |
Colors of variables: wff setvar class |
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) |
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