Step | Hyp | Ref
| Expression |
1 | | otthg 5486 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑))) |
2 | 1 | 3expa 1119 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑))) |
3 | | simp3 1139 |
. . . . . . . . . . 11
⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑) → 𝑐 = 𝑑) |
4 | 2, 3 | syl6bi 253 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑)) |
5 | 4 | con3rr3 155 |
. . . . . . . . 9
⊢ (¬
𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)) |
6 | 5 | imp 408 |
. . . . . . . 8
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩) |
7 | 6 | neqned 2948 |
. . . . . . 7
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩) |
8 | | disjsn2 4717 |
. . . . . . 7
⊢
(⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩ → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅) |
9 | 7, 8 | syl 17 |
. . . . . 6
⊢ ((¬
𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅) |
10 | 9 | expcom 415 |
. . . . 5
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)) |
11 | 10 | orrd 862 |
. . . 4
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)) |
12 | 11 | adantrr 716 |
. . 3
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)) |
13 | 12 | ralrimivva 3201 |
. 2
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)) |
14 | | oteq3 4885 |
. . . 4
⊢ (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩) |
15 | 14 | sneqd 4641 |
. . 3
⊢ (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩}) |
16 | 15 | disjor 5129 |
. 2
⊢
(Disj 𝑐
∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)) |
17 | 13, 16 | sylibr 233 |
1
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩}) |