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 275 | . . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 4 |  | orass 921 | . . . . 5
⊢ ((((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) ∨ ¬ 𝐶 ∈ V) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ (¬ 𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V))) | 
| 5 |  | ianor 983 | . . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴)) | 
| 6 |  | tpprceq3 4803 | . . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴}) | 
| 7 | 5, 6 | sylbir 235 | . . . . . . 7
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴}) | 
| 8 |  | tpcoma 4749 | . . . . . . 7
⊢ {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶} | 
| 9 |  | prcom 4731 | . . . . . . 7
⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | 
| 10 | 7, 8, 9 | 3eqtr3g 2799 | . . . . . 6
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 11 |  | orcom 870 | . . . . . . . 8
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 12 |  | ianor 983 | . . . . . . . 8
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 13 | 11, 12 | bitr4i 278 | . . . . . . 7
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) ↔ ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵)) | 
| 14 |  | tpprceq3 4803 | . . . . . . 7
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 15 | 13, 14 | sylbi 217 | . . . . . 6
⊢ ((¬
𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 16 | 10, 15 | jaoi 857 | . . . . 5
⊢ (((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ (¬ 𝐶 ≠ 𝐵 ∨ ¬ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 17 | 4, 16 | sylbi 217 | . . . 4
⊢ ((((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 18 | 17 | orcs 875 | . . 3
⊢ (((¬
𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴) ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 19 | 3, 18 | sylbi 217 | . 2
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | 
| 20 |  | df-tp 4630 | . . . 4
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | 
| 21 | 20 | eqeq1i 2741 | . . 3
⊢ ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵}) | 
| 22 |  | ssequn2 4188 | . . . 4
⊢ ({𝐶} ⊆ {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵}) | 
| 23 |  | snssg 4782 | . . . . . . 7
⊢ (𝐶 ∈ V → (𝐶 ∈ {𝐴, 𝐵} ↔ {𝐶} ⊆ {𝐴, 𝐵})) | 
| 24 |  | elpri 4648 | . . . . . . . 8
⊢ (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) | 
| 25 |  | nne 2943 | . . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴) | 
| 26 |  | 3mix2 1331 | . . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 27 | 25, 26 | sylbir 235 | . . . . . . . . 9
⊢ (𝐶 = 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 28 |  | nne 2943 | . . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵) | 
| 29 |  | 3mix3 1332 | . . . . . . . . . 10
⊢ (¬
𝐶 ≠ 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 30 | 28, 29 | sylbir 235 | . . . . . . . . 9
⊢ (𝐶 = 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵)) | 
| 31 | 27, 30 | jaoi 857 | . . . . . . . 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 | biimtrrdi 254 | . . . . . 6
⊢ (𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐴 ∨ ¬ 𝐶 ≠ 𝐵))) | 
| 34 |  | 3mix1 1330 | . . . . . . 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 234 | . . . 4
⊢ ({𝐶} ⊆ {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) | 
| 38 | 22, 37 | sylbir 235 | . . 3
⊢ (({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) | 
| 39 | 21, 38 | sylbi 217 | . 2
⊢ ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵)) | 
| 40 | 19, 39 | impbii 209 | 1
⊢ (¬
(𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |