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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 47179 | . . . 4 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
2 | 1 | relopabiv 5828 | . . 3 ⊢ Rel _∉ |
3 | 2 | brrelex12i 5739 | . 2 ⊢ (𝐴 _∉ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | nelbr 47181 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)) | |
5 | 4 | biimpd 229 | . 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 395 ∈ wcel 2104 Vcvv 3477 class class class wbr 5150 _∉ cnelbr 47178 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5151 df-opab 5213 df-xp 5690 df-rel 5691 df-nelbr 47179 |
This theorem is referenced by: nelbrnelim 47184 |
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