Step | Hyp | Ref
| Expression |
1 | | orc 864 |
. . . . 5
⊢ (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)) |
2 | 1 | a1d 25 |
. . . 4
⊢ (𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))) |
3 | | eliun 5001 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ 𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) |
4 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑎 ∈ 𝑉) |
5 | 4 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑎 ∈ 𝑉) |
6 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝐵 ∈ 𝑋) |
7 | | simpl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑐 ∈ 𝑊) |
8 | | otthg 5485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ 𝑊) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒))) |
9 | 5, 6, 7, 8 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒))) |
10 | | simp1 1135 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒) → 𝑎 = 𝑑) |
11 | 9, 10 | syl6bi 253 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑)) |
12 | 11 | con3d 152 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
13 | 12 | ex 412 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 ∈ 𝑊 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))) |
14 | 13 | com13 88 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 ∈ 𝑊 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))) |
15 | 14 | imp31 417 |
. . . . . . . . . . . . . . . . 17
⊢ (((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩) |
16 | 15 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩) |
17 | 16 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩) |
18 | | velsn 4644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩) |
19 | | eqeq1 2735 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
20 | 19 | notbid 318 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
21 | 18, 20 | sylbi 216 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
22 | 21 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
23 | 22 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)) |
24 | 17, 23 | mpbird 257 |
. . . . . . . . . . . . . 14
⊢
(((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩) |
25 | | velsn 4644 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩) |
26 | 24, 25 | sylnibr 329 |
. . . . . . . . . . . . 13
⊢
(((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩}) |
27 | 26 | nrexdv 3148 |
. . . . . . . . . . . 12
⊢ ((((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩}) |
28 | | eliun 5001 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩}) |
29 | 27, 28 | sylnibr 329 |
. . . . . . . . . . 11
⊢ ((((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}) |
30 | 29 | rexlimdva2 3156 |
. . . . . . . . . 10
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (∃𝑐 ∈ 𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})) |
31 | 3, 30 | biimtrid 241 |
. . . . . . . . 9
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})) |
32 | 31 | ralrimiv 3144 |
. . . . . . . 8
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}) |
33 | | oteq3 4884 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩) |
34 | 33 | sneqd 4640 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩}) |
35 | 34 | cbviunv 5043 |
. . . . . . . . . . 11
⊢ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} = ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩} |
36 | 35 | eleq2i 2824 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}) |
37 | 36 | notbii 320 |
. . . . . . . . 9
⊢ (¬
𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}) |
38 | 37 | ralbii 3092 |
. . . . . . . 8
⊢
(∀𝑠 ∈
∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ∀𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪
𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}) |
39 | 32, 38 | sylibr 233 |
. . . . . . 7
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) |
40 | | disj 4447 |
. . . . . . 7
⊢
((∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪
𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) |
41 | 39, 40 | sylibr 233 |
. . . . . 6
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅) |
42 | 41 | olcd 871 |
. . . . 5
⊢ ((¬
𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)) |
43 | 42 | ex 412 |
. . . 4
⊢ (¬
𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))) |
44 | 2, 43 | pm2.61i 182 |
. . 3
⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)) |
45 | 44 | ralrimivva 3199 |
. 2
⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)) |
46 | | oteq1 4882 |
. . . . 5
⊢ (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩) |
47 | 46 | sneqd 4640 |
. . . 4
⊢ (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩}) |
48 | 47 | iuneq2d 5026 |
. . 3
⊢ (𝑎 = 𝑑 → ∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} = ∪
𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) |
49 | 48 | disjor 5128 |
. 2
⊢
(Disj 𝑎
∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪
𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)) |
50 | 45, 49 | sylibr 233 |
1
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩}) |