Proof of Theorem tppreqb
Step | Hyp | Ref
| Expression |
1 | | 3ianor 1106 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
2 | | df-3or 1087 |
. . . 4
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵)) |
3 | 1, 2 | bitri 274 |
. . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵)) |
4 | | orass 919 |
. . . . 5
⊢ ((((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) ∨ ¬ 𝐶 ∈ V) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ (¬ 𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V))) |
5 | | ianor 979 |
. . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴)) |
6 | | tpprceq3 4737 |
. . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴}) |
7 | 5, 6 | sylbir 234 |
. . . . . . 7
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴}) |
8 | | tpcoma 4686 |
. . . . . . 7
⊢ {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶} |
9 | | prcom 4668 |
. . . . . . 7
⊢ {𝐵, 𝐴} = {𝐴, 𝐵} |
10 | 7, 8, 9 | 3eqtr3g 2801 |
. . . . . 6
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
11 | | orcom 867 |
. . . . . . . 8
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) |
12 | | ianor 979 |
. . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) |
13 | 11, 12 | bitr4i 277 |
. . . . . . 7
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) ↔ ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵)) |
14 | | tpprceq3 4737 |
. . . . . . 7
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
15 | 13, 14 | sylbi 216 |
. . . . . 6
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
16 | 10, 15 | jaoi 854 |
. . . . 5
⊢ (((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ (¬ 𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
17 | 4, 16 | sylbi 216 |
. . . 4
⊢ ((((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
18 | 17 | orcs 872 |
. . 3
⊢ (((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
19 | 3, 18 | sylbi 216 |
. 2
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
20 | | df-tp 4566 |
. . . 4
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) |
21 | 20 | eqeq1i 2743 |
. . 3
⊢ ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵}) |
22 | | ssequn2 4117 |
. . . 4
⊢ ({𝐶} ⊆ {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵}) |
23 | | snssg 4718 |
. . . . . . 7
⊢ (𝐶 ∈ V → (𝐶 ∈ {𝐴, 𝐵} ↔ {𝐶} ⊆ {𝐴, 𝐵})) |
24 | | elpri 4583 |
. . . . . . . 8
⊢ (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) |
25 | | nne 2947 |
. . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴) |
26 | | 3mix2 1330 |
. . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
27 | 25, 26 | sylbir 234 |
. . . . . . . . 9
⊢ (𝐶 = 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
28 | | nne 2947 |
. . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵) |
29 | | 3mix3 1331 |
. . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
30 | 28, 29 | sylbir 234 |
. . . . . . . . 9
⊢ (𝐶 = 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
31 | 27, 30 | jaoi 854 |
. . . . . . . 8
⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
32 | 24, 31 | syl 17 |
. . . . . . 7
⊢ (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
33 | 23, 32 | syl6bir 253 |
. . . . . 6
⊢ (𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵))) |
34 | | 3mix1 1329 |
. . . . . . 7
⊢ (¬
𝐶 ∈ V → (¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
35 | 34 | a1d 25 |
. . . . . 6
⊢ (¬
𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵))) |
36 | 33, 35 | pm2.61i 182 |
. . . . 5
⊢ ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) |
37 | 36, 1 | sylibr 233 |
. . . 4
⊢ ({𝐶} ⊆ {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) |
38 | 22, 37 | sylbir 234 |
. . 3
⊢ (({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) |
39 | 21, 38 | sylbi 216 |
. 2
⊢ ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) |
40 | 19, 39 | impbii 208 |
1
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |