| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | lbsext.k | . . . 4
⊢ (𝜑 → 𝒫 𝑉 ∈ dom
card) | 
| 2 |  | lbsext.s | . . . . 5
⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} | 
| 3 | 2 | ssrab3 4081 | . . . 4
⊢ 𝑆 ⊆ 𝒫 𝑉 | 
| 4 |  | ssnum 10080 | . . . 4
⊢
((𝒫 𝑉 ∈
dom card ∧ 𝑆 ⊆
𝒫 𝑉) → 𝑆 ∈ dom
card) | 
| 5 | 1, 3, 4 | sylancl 586 | . . 3
⊢ (𝜑 → 𝑆 ∈ dom card) | 
| 6 |  | lbsext.v | . . . 4
⊢ 𝑉 = (Base‘𝑊) | 
| 7 |  | lbsext.j | . . . 4
⊢ 𝐽 = (LBasis‘𝑊) | 
| 8 |  | lbsext.n | . . . 4
⊢ 𝑁 = (LSpan‘𝑊) | 
| 9 |  | lbsext.w | . . . 4
⊢ (𝜑 → 𝑊 ∈ LVec) | 
| 10 |  | lbsext.c | . . . 4
⊢ (𝜑 → 𝐶 ⊆ 𝑉) | 
| 11 |  | lbsext.x | . . . 4
⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) | 
| 12 | 6, 7, 8, 9, 10, 11, 2 | lbsextlem1 21161 | . . 3
⊢ (𝜑 → 𝑆 ≠ ∅) | 
| 13 | 9 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → 𝑊 ∈ LVec) | 
| 14 | 10 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → 𝐶 ⊆ 𝑉) | 
| 15 | 11 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) | 
| 16 |  | eqid 2736 | . . . . . 6
⊢
(LSubSp‘𝑊) =
(LSubSp‘𝑊) | 
| 17 |  | simpr1 1194 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → 𝑦 ⊆ 𝑆) | 
| 18 |  | simpr2 1195 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → 𝑦 ≠ ∅) | 
| 19 |  | simpr3 1196 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) →
[⊊] Or 𝑦) | 
| 20 |  | eqid 2736 | . . . . . 6
⊢ ∪ 𝑢 ∈ 𝑦 (𝑁‘(𝑢 ∖ {𝑥})) = ∪
𝑢 ∈ 𝑦 (𝑁‘(𝑢 ∖ {𝑥})) | 
| 21 | 6, 7, 8, 13, 14, 15, 2, 16, 17, 18, 19, 20 | lbsextlem3 21163 | . . . . 5
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦)) → ∪ 𝑦
∈ 𝑆) | 
| 22 | 21 | ex 412 | . . . 4
⊢ (𝜑 → ((𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦) → ∪ 𝑦
∈ 𝑆)) | 
| 23 | 22 | alrimiv 1926 | . . 3
⊢ (𝜑 → ∀𝑦((𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦) → ∪ 𝑦
∈ 𝑆)) | 
| 24 |  | zornn0g 10546 | . . 3
⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅ ∧
∀𝑦((𝑦 ⊆ 𝑆 ∧ 𝑦 ≠ ∅ ∧ [⊊] Or
𝑦) → ∪ 𝑦
∈ 𝑆)) →
∃𝑠 ∈ 𝑆 ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡) | 
| 25 | 5, 12, 23, 24 | syl3anc 1372 | . 2
⊢ (𝜑 → ∃𝑠 ∈ 𝑆 ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡) | 
| 26 |  | simprl 770 | . . . . . 6
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑠 ∈ 𝑆) | 
| 27 |  | sseq2 4009 | . . . . . . . 8
⊢ (𝑧 = 𝑠 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝑠)) | 
| 28 |  | difeq1 4118 | . . . . . . . . . . . 12
⊢ (𝑧 = 𝑠 → (𝑧 ∖ {𝑥}) = (𝑠 ∖ {𝑥})) | 
| 29 | 28 | fveq2d 6909 | . . . . . . . . . . 11
⊢ (𝑧 = 𝑠 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥}))) | 
| 30 | 29 | eleq2d 2826 | . . . . . . . . . 10
⊢ (𝑧 = 𝑠 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 31 | 30 | notbid 318 | . . . . . . . . 9
⊢ (𝑧 = 𝑠 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 32 | 31 | raleqbi1dv 3337 | . . . . . . . 8
⊢ (𝑧 = 𝑠 → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 33 | 27, 32 | anbi12d 632 | . . . . . . 7
⊢ (𝑧 = 𝑠 → ((𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶 ⊆ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))) | 
| 34 | 33, 2 | elrab2 3694 | . . . . . 6
⊢ (𝑠 ∈ 𝑆 ↔ (𝑠 ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))) | 
| 35 | 26, 34 | sylib 218 | . . . . 5
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑠 ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))) | 
| 36 | 35 | simpld 494 | . . . 4
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑠 ∈ 𝒫 𝑉) | 
| 37 | 36 | elpwid 4608 | . . 3
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑠 ⊆ 𝑉) | 
| 38 |  | lveclmod 21106 | . . . . . . 7
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | 
| 39 | 9, 38 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝑊 ∈ LMod) | 
| 40 | 39 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑊 ∈ LMod) | 
| 41 | 6, 8 | lspssv 20982 | . . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝑠 ⊆ 𝑉) → (𝑁‘𝑠) ⊆ 𝑉) | 
| 42 | 40, 37, 41 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑁‘𝑠) ⊆ 𝑉) | 
| 43 |  | ssun1 4177 | . . . . . . . . 9
⊢ 𝑠 ⊆ (𝑠 ∪ {𝑤}) | 
| 44 | 43 | a1i 11 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑠 ⊆ (𝑠 ∪ {𝑤})) | 
| 45 |  | ssun2 4178 | . . . . . . . . . . 11
⊢ {𝑤} ⊆ (𝑠 ∪ {𝑤}) | 
| 46 |  | vsnid 4662 | . . . . . . . . . . 11
⊢ 𝑤 ∈ {𝑤} | 
| 47 | 45, 46 | sselii 3979 | . . . . . . . . . 10
⊢ 𝑤 ∈ (𝑠 ∪ {𝑤}) | 
| 48 | 6, 8 | lspssid 20984 | . . . . . . . . . . . . 13
⊢ ((𝑊 ∈ LMod ∧ 𝑠 ⊆ 𝑉) → 𝑠 ⊆ (𝑁‘𝑠)) | 
| 49 | 40, 37, 48 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑠 ⊆ (𝑁‘𝑠)) | 
| 50 | 49 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑠 ⊆ (𝑁‘𝑠)) | 
| 51 |  | eldifn 4131 | . . . . . . . . . . . 12
⊢ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) → ¬ 𝑤 ∈ (𝑁‘𝑠)) | 
| 52 | 51 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ¬ 𝑤 ∈ (𝑁‘𝑠)) | 
| 53 | 50, 52 | ssneldd 3985 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ¬ 𝑤 ∈ 𝑠) | 
| 54 |  | nelne1 3038 | . . . . . . . . . 10
⊢ ((𝑤 ∈ (𝑠 ∪ {𝑤}) ∧ ¬ 𝑤 ∈ 𝑠) → (𝑠 ∪ {𝑤}) ≠ 𝑠) | 
| 55 | 47, 53, 54 | sylancr 587 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑠 ∪ {𝑤}) ≠ 𝑠) | 
| 56 | 55 | necomd 2995 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑠 ≠ (𝑠 ∪ {𝑤})) | 
| 57 |  | df-pss 3970 | . . . . . . . 8
⊢ (𝑠 ⊊ (𝑠 ∪ {𝑤}) ↔ (𝑠 ⊆ (𝑠 ∪ {𝑤}) ∧ 𝑠 ≠ (𝑠 ∪ {𝑤}))) | 
| 58 | 44, 56, 57 | sylanbrc 583 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑠 ⊊ (𝑠 ∪ {𝑤})) | 
| 59 |  | psseq2 4090 | . . . . . . . . 9
⊢ (𝑡 = (𝑠 ∪ {𝑤}) → (𝑠 ⊊ 𝑡 ↔ 𝑠 ⊊ (𝑠 ∪ {𝑤}))) | 
| 60 | 59 | notbid 318 | . . . . . . . 8
⊢ (𝑡 = (𝑠 ∪ {𝑤}) → (¬ 𝑠 ⊊ 𝑡 ↔ ¬ 𝑠 ⊊ (𝑠 ∪ {𝑤}))) | 
| 61 |  | simplrr 777 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡) | 
| 62 | 37 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑠 ⊆ 𝑉) | 
| 63 |  | eldifi 4130 | . . . . . . . . . . . . 13
⊢ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) → 𝑤 ∈ 𝑉) | 
| 64 | 63 | adantl 481 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝑤 ∈ 𝑉) | 
| 65 | 64 | snssd 4808 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → {𝑤} ⊆ 𝑉) | 
| 66 | 62, 65 | unssd 4191 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑠 ∪ {𝑤}) ⊆ 𝑉) | 
| 67 | 6 | fvexi 6919 | . . . . . . . . . . 11
⊢ 𝑉 ∈ V | 
| 68 | 67 | elpw2 5333 | . . . . . . . . . 10
⊢ ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑉 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑉) | 
| 69 | 66, 68 | sylibr 234 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑠 ∪ {𝑤}) ∈ 𝒫 𝑉) | 
| 70 | 35 | simprd 495 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝐶 ⊆ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 71 | 70 | simpld 494 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝐶 ⊆ 𝑠) | 
| 72 | 71 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝐶 ⊆ 𝑠) | 
| 73 | 72, 43 | sstrdi 3995 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → 𝐶 ⊆ (𝑠 ∪ {𝑤})) | 
| 74 | 9 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑊 ∈ LVec) | 
| 75 | 37 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑠 ⊆ 𝑉) | 
| 76 | 75 | ssdifssd 4146 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → (𝑠 ∖ {𝑥}) ⊆ 𝑉) | 
| 77 | 64 | adantrr 717 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑤 ∈ 𝑉) | 
| 78 |  | simprrr 781 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 79 |  | difundir 4290 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∪ {𝑤}) ∖ {𝑥}) = ((𝑠 ∖ {𝑥}) ∪ ({𝑤} ∖ {𝑥})) | 
| 80 |  | simprrl 780 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑥 ∈ 𝑠) | 
| 81 | 53 | adantrr 717 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ¬ 𝑤 ∈ 𝑠) | 
| 82 |  | nelne2 3039 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ 𝑠 ∧ ¬ 𝑤 ∈ 𝑠) → 𝑥 ≠ 𝑤) | 
| 83 | 80, 81, 82 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑥 ≠ 𝑤) | 
| 84 |  | nelsn 4665 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ≠ 𝑤 → ¬ 𝑥 ∈ {𝑤}) | 
| 85 | 83, 84 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ¬ 𝑥 ∈ {𝑤}) | 
| 86 |  | disjsn 4710 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (({𝑤} ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ {𝑤}) | 
| 87 | 85, 86 | sylibr 234 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ({𝑤} ∩ {𝑥}) = ∅) | 
| 88 |  | disj3 4453 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({𝑤} ∩ {𝑥}) = ∅ ↔ {𝑤} = ({𝑤} ∖ {𝑥})) | 
| 89 | 87, 88 | sylib 218 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → {𝑤} = ({𝑤} ∖ {𝑥})) | 
| 90 | 89 | uneq2d 4167 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ((𝑠 ∖ {𝑥}) ∪ {𝑤}) = ((𝑠 ∖ {𝑥}) ∪ ({𝑤} ∖ {𝑥}))) | 
| 91 | 79, 90 | eqtr4id 2795 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ((𝑠 ∪ {𝑤}) ∖ {𝑥}) = ((𝑠 ∖ {𝑥}) ∪ {𝑤})) | 
| 92 | 91 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) = (𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑤}))) | 
| 93 | 78, 92 | eleqtrd 2842 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑥 ∈ (𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑤}))) | 
| 94 | 70 | simprd 495 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))) | 
| 95 | 94 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))) | 
| 96 |  | rsp 3246 | . . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑥 ∈
𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) → (𝑥 ∈ 𝑠 → ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 97 | 95, 80, 96 | sylc 65 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))) | 
| 98 | 93, 97 | eldifd 3961 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑥 ∈ ((𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑤})) ∖ (𝑁‘(𝑠 ∖ {𝑥})))) | 
| 99 | 6, 16, 8 | lspsolv 21146 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑊 ∈ LVec ∧ ((𝑠 ∖ {𝑥}) ⊆ 𝑉 ∧ 𝑤 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑤})) ∖ (𝑁‘(𝑠 ∖ {𝑥}))))) → 𝑤 ∈ (𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑥}))) | 
| 100 | 74, 76, 77, 98, 99 | syl13anc 1373 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑤 ∈ (𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑥}))) | 
| 101 |  | undif1 4475 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∖ {𝑥}) ∪ {𝑥}) = (𝑠 ∪ {𝑥}) | 
| 102 | 80 | snssd 4808 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → {𝑥} ⊆ 𝑠) | 
| 103 |  | ssequn2 4188 | . . . . . . . . . . . . . . . . . . 19
⊢ ({𝑥} ⊆ 𝑠 ↔ (𝑠 ∪ {𝑥}) = 𝑠) | 
| 104 | 102, 103 | sylib 218 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → (𝑠 ∪ {𝑥}) = 𝑠) | 
| 105 | 101, 104 | eqtrid 2788 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → ((𝑠 ∖ {𝑥}) ∪ {𝑥}) = 𝑠) | 
| 106 | 105 | fveq2d 6909 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → (𝑁‘((𝑠 ∖ {𝑥}) ∪ {𝑥})) = (𝑁‘𝑠)) | 
| 107 | 100, 106 | eleqtrd 2842 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ (𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) → 𝑤 ∈ (𝑁‘𝑠)) | 
| 108 | 107 | expr 456 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ((𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) → 𝑤 ∈ (𝑁‘𝑠))) | 
| 109 | 52, 108 | mtod 198 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ¬ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 110 |  | imnan 399 | . . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑠 → ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) ↔ ¬ (𝑥 ∈ 𝑠 ∧ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 111 | 109, 110 | sylibr 234 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑥 ∈ 𝑠 → ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 112 | 111 | ralrimiv 3144 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 113 |  | difssd 4136 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑠 ∖ {𝑤}) ⊆ 𝑠) | 
| 114 | 6, 8 | lspss 20983 | . . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ LMod ∧ 𝑠 ⊆ 𝑉 ∧ (𝑠 ∖ {𝑤}) ⊆ 𝑠) → (𝑁‘(𝑠 ∖ {𝑤})) ⊆ (𝑁‘𝑠)) | 
| 115 | 40, 37, 113, 114 | syl3anc 1372 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑁‘(𝑠 ∖ {𝑤})) ⊆ (𝑁‘𝑠)) | 
| 116 | 115 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑁‘(𝑠 ∖ {𝑤})) ⊆ (𝑁‘𝑠)) | 
| 117 | 116, 52 | ssneldd 3985 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ¬ 𝑤 ∈ (𝑁‘(𝑠 ∖ {𝑤}))) | 
| 118 |  | vex 3483 | . . . . . . . . . . . . 13
⊢ 𝑤 ∈ V | 
| 119 |  | id 22 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤) | 
| 120 |  | sneq 4635 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤}) | 
| 121 | 120 | difeq2d 4125 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑤 → ((𝑠 ∪ {𝑤}) ∖ {𝑥}) = ((𝑠 ∪ {𝑤}) ∖ {𝑤})) | 
| 122 |  | difun2 4480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∪ {𝑤}) ∖ {𝑤}) = (𝑠 ∖ {𝑤}) | 
| 123 | 121, 122 | eqtrdi 2792 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑤 → ((𝑠 ∪ {𝑤}) ∖ {𝑥}) = (𝑠 ∖ {𝑤})) | 
| 124 | 123 | fveq2d 6909 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑤 → (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑤}))) | 
| 125 | 119, 124 | eleq12d 2834 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑤 → (𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑠 ∖ {𝑤})))) | 
| 126 | 125 | notbid 318 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑤 → (¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) ↔ ¬ 𝑤 ∈ (𝑁‘(𝑠 ∖ {𝑤})))) | 
| 127 | 118, 126 | ralsn 4680 | . . . . . . . . . . . 12
⊢
(∀𝑥 ∈
{𝑤} ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) ↔ ¬ 𝑤 ∈ (𝑁‘(𝑠 ∖ {𝑤}))) | 
| 128 | 117, 127 | sylibr 234 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ∀𝑥 ∈ {𝑤} ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 129 |  | ralun 4197 | . . . . . . . . . . 11
⊢
((∀𝑥 ∈
𝑠 ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})) ∧ ∀𝑥 ∈ {𝑤} ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) → ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 130 | 112, 128,
129 | syl2anc 584 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 131 | 73, 130 | jca 511 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝐶 ⊆ (𝑠 ∪ {𝑤}) ∧ ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 132 |  | sseq2 4009 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ (𝑠 ∪ {𝑤}))) | 
| 133 |  | difeq1 4118 | . . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (𝑧 ∖ {𝑥}) = ((𝑠 ∪ {𝑤}) ∖ {𝑥})) | 
| 134 | 133 | fveq2d 6909 | . . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))) | 
| 135 | 134 | eleq2d 2826 | . . . . . . . . . . . . 13
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 136 | 135 | notbid 318 | . . . . . . . . . . . 12
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 137 | 136 | raleqbi1dv 3337 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥})))) | 
| 138 | 132, 137 | anbi12d 632 | . . . . . . . . . 10
⊢ (𝑧 = (𝑠 ∪ {𝑤}) → ((𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶 ⊆ (𝑠 ∪ {𝑤}) ∧ ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) | 
| 139 | 138, 2 | elrab2 3694 | . . . . . . . . 9
⊢ ((𝑠 ∪ {𝑤}) ∈ 𝑆 ↔ ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ (𝑠 ∪ {𝑤}) ∧ ∀𝑥 ∈ (𝑠 ∪ {𝑤}) ¬ 𝑥 ∈ (𝑁‘((𝑠 ∪ {𝑤}) ∖ {𝑥}))))) | 
| 140 | 69, 131, 139 | sylanbrc 583 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → (𝑠 ∪ {𝑤}) ∈ 𝑆) | 
| 141 | 60, 61, 140 | rspcdva 3622 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) ∧ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) → ¬ 𝑠 ⊊ (𝑠 ∪ {𝑤})) | 
| 142 | 58, 141 | pm2.65da 816 | . . . . . 6
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → ¬ 𝑤 ∈ (𝑉 ∖ (𝑁‘𝑠))) | 
| 143 | 142 | eq0rdv 4406 | . . . . 5
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑉 ∖ (𝑁‘𝑠)) = ∅) | 
| 144 |  | ssdif0 4365 | . . . . 5
⊢ (𝑉 ⊆ (𝑁‘𝑠) ↔ (𝑉 ∖ (𝑁‘𝑠)) = ∅) | 
| 145 | 143, 144 | sylibr 234 | . . . 4
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑉 ⊆ (𝑁‘𝑠)) | 
| 146 | 42, 145 | eqssd 4000 | . . 3
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑁‘𝑠) = 𝑉) | 
| 147 | 9 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑊 ∈ LVec) | 
| 148 | 6, 7, 8 | islbs2 21157 | . . . 4
⊢ (𝑊 ∈ LVec → (𝑠 ∈ 𝐽 ↔ (𝑠 ⊆ 𝑉 ∧ (𝑁‘𝑠) = 𝑉 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))) | 
| 149 | 147, 148 | syl 17 | . . 3
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → (𝑠 ∈ 𝐽 ↔ (𝑠 ⊆ 𝑉 ∧ (𝑁‘𝑠) = 𝑉 ∧ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))) | 
| 150 | 37, 146, 94, 149 | mpbir3and 1342 | . 2
⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 ¬ 𝑠 ⊊ 𝑡)) → 𝑠 ∈ 𝐽) | 
| 151 | 25, 150, 71 | reximssdv 3172 | 1
⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) |