Step | Hyp | Ref
| Expression |
1 | | orc 866 |
. . . . 5
⊢ (𝑐 = 𝑑 → (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅)) |
2 | 1 | a1d 25 |
. . . 4
⊢ (𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅))) |
3 | | s3cli 14832 |
. . . . . . . . . . . 12
⊢
⟨“𝐴𝐵𝑐”⟩ ∈ Word V |
4 | | elex 3493 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V) |
5 | | elex 3493 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ V) |
6 | 4, 5 | anim12i 614 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | | elex 3493 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 ∈ 𝑍 → 𝑑 ∈ V) |
8 | 7 | adantl 483 |
. . . . . . . . . . . . . 14
⊢ ((𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍) → 𝑑 ∈ V) |
9 | 6, 8 | anim12i 614 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑑 ∈ V)) |
10 | | df-3an 1090 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑑 ∈ V) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑑 ∈ V)) |
11 | 9, 10 | sylibr 233 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑑 ∈ V)) |
12 | | eqwrds3 14912 |
. . . . . . . . . . . 12
⊢
((⟨“𝐴𝐵𝑐”⟩ ∈ Word V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑑 ∈ V)) →
(⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩ ↔
((♯‘⟨“𝐴𝐵𝑐”⟩) = 3 ∧ ((⟨“𝐴𝐵𝑐”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑)))) |
13 | 3, 11, 12 | sylancr 588 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩ ↔
((♯‘⟨“𝐴𝐵𝑐”⟩) = 3 ∧ ((⟨“𝐴𝐵𝑐”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑)))) |
14 | | s3fv2 14844 |
. . . . . . . . . . . . . 14
⊢ (𝑐 ∈ V →
(⟨“𝐴𝐵𝑐”⟩‘2) = 𝑐) |
15 | 14 | elv 3481 |
. . . . . . . . . . . . 13
⊢
(⟨“𝐴𝐵𝑐”⟩‘2) = 𝑐 |
16 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢
(((⟨“𝐴𝐵𝑐”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑) → (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑) |
17 | 15, 16 | eqtr3id 2787 |
. . . . . . . . . . . 12
⊢
(((⟨“𝐴𝐵𝑐”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑) → 𝑐 = 𝑑) |
18 | 17 | adantl 483 |
. . . . . . . . . . 11
⊢
(((♯‘⟨“𝐴𝐵𝑐”⟩) = 3 ∧ ((⟨“𝐴𝐵𝑐”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝐴𝐵𝑐”⟩‘2) = 𝑑)) → 𝑐 = 𝑑) |
19 | 13, 18 | syl6bi 253 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩ → 𝑐 = 𝑑)) |
20 | 19 | con3rr3 155 |
. . . . . . . . 9
⊢ (¬
𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → ¬ ⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩)) |
21 | 20 | imp 408 |
. . . . . . . 8
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → ¬ ⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩) |
22 | 21 | neqned 2948 |
. . . . . . 7
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → ⟨“𝐴𝐵𝑐”⟩ ≠ ⟨“𝐴𝐵𝑑”⟩) |
23 | | disjsn2 4717 |
. . . . . . 7
⊢
(⟨“𝐴𝐵𝑐”⟩ ≠ ⟨“𝐴𝐵𝑑”⟩ → ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅) |
24 | 22, 23 | syl 17 |
. . . . . 6
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅) |
25 | 24 | olcd 873 |
. . . . 5
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅)) |
26 | 25 | ex 414 |
. . . 4
⊢ (¬
𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅))) |
27 | 2, 26 | pm2.61i 182 |
. . 3
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅)) |
28 | 27 | ralrimivva 3201 |
. 2
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ∀𝑐 ∈ 𝑍 ∀𝑑 ∈ 𝑍 (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅)) |
29 | | eqidd 2734 |
. . . . 5
⊢ (𝑐 = 𝑑 → 𝐴 = 𝐴) |
30 | | eqidd 2734 |
. . . . 5
⊢ (𝑐 = 𝑑 → 𝐵 = 𝐵) |
31 | | id 22 |
. . . . 5
⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑) |
32 | 29, 30, 31 | s3eqd 14815 |
. . . 4
⊢ (𝑐 = 𝑑 → ⟨“𝐴𝐵𝑐”⟩ = ⟨“𝐴𝐵𝑑”⟩) |
33 | 32 | sneqd 4641 |
. . 3
⊢ (𝑐 = 𝑑 → {⟨“𝐴𝐵𝑐”⟩} = {⟨“𝐴𝐵𝑑”⟩}) |
34 | 33 | disjor 5129 |
. 2
⊢
(Disj 𝑐
∈ 𝑍
{⟨“𝐴𝐵𝑐”⟩} ↔ ∀𝑐 ∈ 𝑍 ∀𝑑 ∈ 𝑍 (𝑐 = 𝑑 ∨ ({⟨“𝐴𝐵𝑐”⟩} ∩ {⟨“𝐴𝐵𝑑”⟩}) = ∅)) |
35 | 28, 34 | sylibr 233 |
1
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑍 {⟨“𝐴𝐵𝑐”⟩}) |