| Step | Hyp | Ref
| Expression |
| 1 | | islbs2.v |
. . . . 5
⊢ 𝑉 = (Base‘𝑊) |
| 2 | | islbs2.j |
. . . . 5
⊢ 𝐽 = (LBasis‘𝑊) |
| 3 | 1, 2 | lbsss 21076 |
. . . 4
⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) |
| 4 | 3 | adantl 481 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → 𝐵 ⊆ 𝑉) |
| 5 | | islbs2.n |
. . . . 5
⊢ 𝑁 = (LSpan‘𝑊) |
| 6 | 1, 2, 5 | lbssp 21078 |
. . . 4
⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) |
| 7 | 6 | adantl 481 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → (𝑁‘𝐵) = 𝑉) |
| 8 | | lveclmod 21105 |
. . . . . . 7
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) |
| 9 | | eqid 2737 |
. . . . . . . . 9
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
| 10 | 9 | lvecdrng 21104 |
. . . . . . . 8
⊢ (𝑊 ∈ LVec →
(Scalar‘𝑊) ∈
DivRing) |
| 11 | | eqid 2737 |
. . . . . . . . 9
⊢
(0g‘(Scalar‘𝑊)) =
(0g‘(Scalar‘𝑊)) |
| 12 | | eqid 2737 |
. . . . . . . . 9
⊢
(1r‘(Scalar‘𝑊)) =
(1r‘(Scalar‘𝑊)) |
| 13 | 11, 12 | drngunz 20747 |
. . . . . . . 8
⊢
((Scalar‘𝑊)
∈ DivRing → (1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) |
| 14 | 10, 13 | syl 17 |
. . . . . . 7
⊢ (𝑊 ∈ LVec →
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) |
| 15 | 8, 14 | jca 511 |
. . . . . 6
⊢ (𝑊 ∈ LVec → (𝑊 ∈ LMod ∧
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊)))) |
| 16 | 2, 5, 9, 12, 11 | lbsind2 21080 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) ∧ 𝐵 ∈ 𝐽 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
| 17 | 15, 16 | syl3an1 1164 |
. . . . 5
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
| 18 | 17 | 3expa 1119 |
. . . 4
⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
| 19 | 18 | ralrimiva 3146 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
| 20 | 4, 7, 19 | 3jca 1129 |
. 2
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) |
| 21 | | simpr1 1195 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝐵 ⊆ 𝑉) |
| 22 | | simpr2 1196 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → (𝑁‘𝐵) = 𝑉) |
| 23 | | id 22 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
| 24 | | sneq 4636 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) |
| 25 | 24 | difeq2d 4126 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑦})) |
| 26 | 25 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑦}))) |
| 27 | 23, 26 | eleq12d 2835 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 28 | 27 | notbid 318 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 29 | | simplr3 1218 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
| 30 | | simprl 771 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ 𝐵) |
| 31 | 28, 29, 30 | rspcdva 3623 |
. . . . 5
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
| 32 | | simpll 767 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑊 ∈ LVec) |
| 33 | | simprr 773 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))})) |
| 34 | | eldifsn 4786 |
. . . . . . . . 9
⊢ (𝑧 ∈
((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ↔ (𝑧 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊)))) |
| 35 | 33, 34 | sylib 218 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑧 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊)))) |
| 36 | 21 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝐵 ⊆ 𝑉) |
| 37 | 36, 30 | sseldd 3984 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ 𝑉) |
| 38 | | eqid 2737 |
. . . . . . . . 9
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
| 39 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
| 40 | 1, 9, 38, 39, 11, 5 | lspsnvs 21116 |
. . . . . . . 8
⊢ ((𝑊 ∈ LVec ∧ (𝑧 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊))) ∧ 𝑦 ∈ 𝑉) → (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) = (𝑁‘{𝑦})) |
| 41 | 32, 35, 37, 40 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) = (𝑁‘{𝑦})) |
| 42 | 41 | sseq1d 4015 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{𝑦}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 43 | | eqid 2737 |
. . . . . . 7
⊢
(LSubSp‘𝑊) =
(LSubSp‘𝑊) |
| 44 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝑊 ∈ LMod) |
| 45 | 44 | adantr 480 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑊 ∈ LMod) |
| 46 | 36 | ssdifssd 4147 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝐵 ∖ {𝑦}) ⊆ 𝑉) |
| 47 | 1, 43, 5 | lspcl 20974 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ {𝑦}) ⊆ 𝑉) → (𝑁‘(𝐵 ∖ {𝑦})) ∈ (LSubSp‘𝑊)) |
| 48 | 45, 46, 47 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑁‘(𝐵 ∖ {𝑦})) ∈ (LSubSp‘𝑊)) |
| 49 | 35 | simpld 494 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑧 ∈ (Base‘(Scalar‘𝑊))) |
| 50 | 1, 9, 38, 39 | lmodvscl 20876 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉) → (𝑧( ·𝑠
‘𝑊)𝑦) ∈ 𝑉) |
| 51 | 45, 49, 37, 50 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑧( ·𝑠
‘𝑊)𝑦) ∈ 𝑉) |
| 52 | 1, 43, 5, 45, 48, 51 | ellspsn5b 20993 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 53 | 1, 43, 5, 45, 48, 37 | ellspsn5b 20993 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{𝑦}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 54 | 42, 52, 53 | 3bitr4d 311 |
. . . . 5
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
| 55 | 31, 54 | mtbird 325 |
. . . 4
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
| 56 | 55 | ralrimivva 3202 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
| 57 | 1, 9, 38, 39, 2, 5, 11 | islbs 21075 |
. . . 4
⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))))) |
| 58 | 57 | adantr 480 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))))) |
| 59 | 21, 22, 56, 58 | mpbir3and 1343 |
. 2
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝐵 ∈ 𝐽) |
| 60 | 20, 59 | impbida 801 |
1
⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))))) |