Proof of Theorem conndisj
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | connclo.3 | . 2
⊢ (𝜑 → 𝐴 ≠ ∅) | 
| 2 |  | connclo.2 | . . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝐽) | 
| 3 |  | elssuni 4936 | . . . . . . 7
⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | 
| 4 | 2, 3 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) | 
| 5 |  | isconn.1 | . . . . . 6
⊢ 𝑋 = ∪
𝐽 | 
| 6 | 4, 5 | sseqtrrdi 4024 | . . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝑋) | 
| 7 |  | conndisj.6 | . . . . 5
⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | 
| 8 |  | uneqdifeq 4492 | . . . . 5
⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵)) | 
| 9 | 6, 7, 8 | syl2anc 584 | . . . 4
⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵)) | 
| 10 |  | simpr 484 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵) | 
| 11 | 10 | difeq2d 4125 | . . . . . 6
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵)) | 
| 12 |  | dfss4 4268 | . . . . . . . 8
⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | 
| 13 | 6, 12 | sylib 218 | . . . . . . 7
⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | 
| 14 | 13 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | 
| 15 |  | connclo.1 | . . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ Conn) | 
| 16 | 15 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn) | 
| 17 |  | conndisj.4 | . . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ 𝐽) | 
| 18 | 17 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽) | 
| 19 |  | conndisj.5 | . . . . . . . . . 10
⊢ (𝜑 → 𝐵 ≠ ∅) | 
| 20 | 19 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅) | 
| 21 | 5 | isconn 23422 | . . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) | 
| 22 | 21 | simplbi 497 | . . . . . . . . . . . . 13
⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | 
| 23 | 15, 22 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ Top) | 
| 24 | 5 | opncld 23042 | . . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽)) | 
| 25 | 23, 2, 24 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽)) | 
| 26 | 25 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽)) | 
| 27 | 10, 26 | eqeltrrd 2841 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽)) | 
| 28 | 5, 16, 18, 20, 27 | connclo 23424 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋) | 
| 29 | 28 | difeq2d 4125 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋)) | 
| 30 |  | difid 4375 | . . . . . . 7
⊢ (𝑋 ∖ 𝑋) = ∅ | 
| 31 | 29, 30 | eqtrdi 2792 | . . . . . 6
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅) | 
| 32 | 11, 14, 31 | 3eqtr3d 2784 | . . . . 5
⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅) | 
| 33 | 32 | ex 412 | . . . 4
⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅)) | 
| 34 | 9, 33 | sylbid 240 | . . 3
⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅)) | 
| 35 | 34 | necon3d 2960 | . 2
⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋)) | 
| 36 | 1, 35 | mpd 15 | 1
⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋) |