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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelbrim | 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. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴 ∈ 𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.) |
Ref | Expression |
---|---|
nelbrim | ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nelbr 46787 | . . . 4 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
2 | 1 | relopabiv 5822 | . . 3 ⊢ Rel _∉ |
3 | 2 | brrelex12i 5733 | . 2 ⊢ (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | nelbr 46789 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)) | |
5 | 4 | biimpd 228 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)) |
6 | 3, 5 | mpcom 38 | 1 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 class class class wbr 5149 _∉ cnelbr 46786 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-nelbr 46787 |
This theorem is referenced by: nelbrnelim 46792 |
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