| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | pm2.1 897 | . . 3
⊢ (¬
{𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) | 
| 2 |  | df-ne 2941 | . . . . 5
⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ ↔ ¬ {𝐴, 𝐵, 𝐶} = ∅) | 
| 3 | 2 | bicomi 224 | . . . 4
⊢ (¬
{𝐴, 𝐵, 𝐶} = ∅ ↔ {𝐴, 𝐵, 𝐶} ≠ ∅) | 
| 4 | 3 | orbi1i 914 | . . 3
⊢ ((¬
{𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) ↔ ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅)) | 
| 5 | 1, 4 | mpbi 230 | . 2
⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) | 
| 6 |  | zfregs2 9773 | . . . 4
⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) | 
| 7 |  | en3lplem2VD 44864 | . . . . . . 7
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) | 
| 8 | 7 | alrimiv 1927 | . . . . . 6
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) | 
| 9 |  | df-ral 3062 | . . . . . 6
⊢
(∀𝑥 ∈
{𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) | 
| 10 | 8, 9 | sylibr 234 | . . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) | 
| 11 | 10 | con3i 154 | . . . 4
⊢ (¬
∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) | 
| 12 | 6, 11 | syl 17 | . . 3
⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) | 
| 13 |  | idn1 44594 | . . . . . . 7
⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶   {𝐴, 𝐵, 𝐶} = ∅   ) | 
| 14 |  | noel 4338 | . . . . . . 7
⊢  ¬
𝐶 ∈
∅ | 
| 15 |  | eleq2 2830 | . . . . . . . . 9
⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅)) | 
| 16 | 15 | notbid 318 | . . . . . . . 8
⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ ¬ 𝐶 ∈ ∅)) | 
| 17 | 16 | biimprd 248 | . . . . . . 7
⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) | 
| 18 | 13, 14, 17 | e10 44714 | . . . . . 6
⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬
𝐶 ∈ {𝐴, 𝐵, 𝐶}   ) | 
| 19 |  | tpid3g 4772 | . . . . . . 7
⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶}) | 
| 20 | 19 | con3i 154 | . . . . . 6
⊢ (¬
𝐶 ∈ {𝐴, 𝐵, 𝐶} → ¬ 𝐶 ∈ 𝐴) | 
| 21 | 18, 20 | e1a 44647 | . . . . 5
⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬
𝐶 ∈ 𝐴   ) | 
| 22 |  | simp3 1139 | . . . . . 6
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴) | 
| 23 | 22 | con3i 154 | . . . . 5
⊢ (¬
𝐶 ∈ 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) | 
| 24 | 21, 23 | e1a 44647 | . . . 4
⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬
(𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ) | 
| 25 | 24 | in1 44591 | . . 3
⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) | 
| 26 | 12, 25 | jaoi 858 | . 2
⊢ (({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) | 
| 27 | 5, 26 | ax-mp 5 | 1
⊢  ¬
(𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |