| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | orc 867 | . . . . 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 14921 | . . . . . . . . . . . 12
⊢
〈“𝐴𝐵𝑐”〉 ∈ Word V | 
| 4 |  | elex 3500 | . . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V) | 
| 5 |  | elex 3500 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ V) | 
| 6 | 4, 5 | anim12i 613 | . . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| 7 |  | elex 3500 | . . . . . . . . . . . . . . 15
⊢ (𝑑 ∈ 𝑍 → 𝑑 ∈ V) | 
| 8 | 7 | adantl 481 | . . . . . . . . . . . . . 14
⊢ ((𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍) → 𝑑 ∈ V) | 
| 9 | 6, 8 | anim12i 613 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑑 ∈ V)) | 
| 10 |  | df-3an 1088 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑑 ∈ V) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑑 ∈ V)) | 
| 11 | 9, 10 | sylibr 234 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑑 ∈ V)) | 
| 12 |  | eqwrds3 15001 | . . . . . . . . . . . 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 587 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍)) → (〈“𝐴𝐵𝑐”〉 = 〈“𝐴𝐵𝑑”〉 ↔
((♯‘〈“𝐴𝐵𝑐”〉) = 3 ∧ ((〈“𝐴𝐵𝑐”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝑐”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝑐”〉‘2) = 𝑑)))) | 
| 14 |  | s3fv2 14933 | . . . . . . . . . . . . . 14
⊢ (𝑐 ∈ V →
(〈“𝐴𝐵𝑐”〉‘2) = 𝑐) | 
| 15 | 14 | elv 3484 | . . . . . . . . . . . . 13
⊢
(〈“𝐴𝐵𝑐”〉‘2) = 𝑐 | 
| 16 |  | simp3 1138 | . . . . . . . . . . . . 13
⊢
(((〈“𝐴𝐵𝑐”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝑐”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝑐”〉‘2) = 𝑑) → (〈“𝐴𝐵𝑐”〉‘2) = 𝑑) | 
| 17 | 15, 16 | eqtr3id 2790 | . . . . . . . . . . . 12
⊢
(((〈“𝐴𝐵𝑐”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝑐”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝑐”〉‘2) = 𝑑) → 𝑐 = 𝑑) | 
| 18 | 17 | adantl 481 | . . . . . . . . . . 11
⊢
(((♯‘〈“𝐴𝐵𝑐”〉) = 3 ∧ ((〈“𝐴𝐵𝑐”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝑐”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝑐”〉‘2) = 𝑑)) → 𝑐 = 𝑑) | 
| 19 | 13, 18 | biimtrdi 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 406 | . . . . . . . 8
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → ¬ 〈“𝐴𝐵𝑐”〉 = 〈“𝐴𝐵𝑑”〉) | 
| 22 | 21 | neqned 2946 | . . . . . . 7
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → 〈“𝐴𝐵𝑐”〉 ≠ 〈“𝐴𝐵𝑑”〉) | 
| 23 |  | disjsn2 4711 | . . . . . . 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 874 | . . . . 5
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑍 ∧ 𝑑 ∈ 𝑍))) → (𝑐 = 𝑑 ∨ ({〈“𝐴𝐵𝑐”〉} ∩ {〈“𝐴𝐵𝑑”〉}) = ∅)) | 
| 26 | 25 | ex 412 | . . . 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 2737 | . . . . 5
⊢ (𝑐 = 𝑑 → 𝐴 = 𝐴) | 
| 30 |  | eqidd 2737 | . . . . 5
⊢ (𝑐 = 𝑑 → 𝐵 = 𝐵) | 
| 31 |  | id 22 | . . . . 5
⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑) | 
| 32 | 29, 30, 31 | s3eqd 14904 | . . . 4
⊢ (𝑐 = 𝑑 → 〈“𝐴𝐵𝑐”〉 = 〈“𝐴𝐵𝑑”〉) | 
| 33 | 32 | sneqd 4637 | . . 3
⊢ (𝑐 = 𝑑 → {〈“𝐴𝐵𝑐”〉} = {〈“𝐴𝐵𝑑”〉}) | 
| 34 | 33 | disjor 5124 | . 2
⊢
(Disj 𝑐
∈ 𝑍
{〈“𝐴𝐵𝑐”〉} ↔ ∀𝑐 ∈ 𝑍 ∀𝑑 ∈ 𝑍 (𝑐 = 𝑑 ∨ ({〈“𝐴𝐵𝑐”〉} ∩ {〈“𝐴𝐵𝑑”〉}) = ∅)) | 
| 35 | 28, 34 | sylibr 234 | 1
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑍 {〈“𝐴𝐵𝑐”〉}) |