Proof of Theorem regsep2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | regtop 23342 | . . . . . . 7
⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | 
| 2 | 1 | ad2antrr 726 | . . . . . 6
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐽 ∈ Top) | 
| 3 |  | elssuni 4936 | . . . . . . . 8
⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽) | 
| 4 |  | t1sep.1 | . . . . . . . 8
⊢ 𝑋 = ∪
𝐽 | 
| 5 | 3, 4 | sseqtrrdi 4024 | . . . . . . 7
⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ 𝑋) | 
| 6 | 5 | ad2antrl 728 | . . . . . 6
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝑦 ⊆ 𝑋) | 
| 7 | 4 | clscld 23056 | . . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) | 
| 8 | 2, 6, 7 | syl2anc 584 | . . . . 5
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) | 
| 9 | 4 | cldopn 23040 | . . . . 5
⊢
(((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽) | 
| 10 | 8, 9 | syl 17 | . . . 4
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽) | 
| 11 |  | simprrr 781 | . . . . 5
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)) | 
| 12 | 4 | clsss3 23068 | . . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋) | 
| 13 | 2, 6, 12 | syl2anc 584 | . . . . . 6
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋) | 
| 14 |  | simplr1 1215 | . . . . . . 7
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ∈ (Clsd‘𝐽)) | 
| 15 | 4 | cldss 23038 | . . . . . . 7
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋) | 
| 16 | 14, 15 | syl 17 | . . . . . 6
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ⊆ 𝑋) | 
| 17 |  | ssconb 4141 | . . . . . 6
⊢
((((cls‘𝐽)‘𝑦) ⊆ 𝑋 ∧ 𝐶 ⊆ 𝑋) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))) | 
| 18 | 13, 16, 17 | syl2anc 584 | . . . . 5
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))) | 
| 19 | 11, 18 | mpbid 232 | . . . 4
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))) | 
| 20 |  | simprrl 780 | . . . 4
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐴 ∈ 𝑦) | 
| 21 | 4 | sscls 23065 | . . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦)) | 
| 22 | 2, 6, 21 | syl2anc 584 | . . . . . 6
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦)) | 
| 23 |  | sslin 4242 | . . . . . 6
⊢ (𝑦 ⊆ ((cls‘𝐽)‘𝑦) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦))) | 
| 24 | 22, 23 | syl 17 | . . . . 5
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦))) | 
| 25 |  | disjdifr 4472 | . . . . 5
⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅ | 
| 26 |  | sseq0 4402 | . . . . 5
⊢ ((((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅) | 
| 27 | 24, 25, 26 | sylancl 586 | . . . 4
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅) | 
| 28 |  | sseq2 4009 | . . . . . 6
⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶 ⊆ 𝑥 ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))) | 
| 29 |  | ineq1 4212 | . . . . . . 7
⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥 ∩ 𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦)) | 
| 30 | 29 | eqeq1d 2738 | . . . . . 6
⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)) | 
| 31 | 28, 30 | 3anbi13d 1439 | . . . . 5
⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴 ∈ 𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))) | 
| 32 | 31 | rspcev 3621 | . . . 4
⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽 ∧ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴 ∈ 𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | 
| 33 | 10, 19, 20, 27, 32 | syl13anc 1373 | . . 3
⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | 
| 34 |  | simpl 482 | . . . 4
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐽 ∈ Reg) | 
| 35 |  | simpr1 1194 | . . . . 5
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ (Clsd‘𝐽)) | 
| 36 | 4 | cldopn 23040 | . . . . 5
⊢ (𝐶 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐶) ∈ 𝐽) | 
| 37 | 35, 36 | syl 17 | . . . 4
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → (𝑋 ∖ 𝐶) ∈ 𝐽) | 
| 38 |  | simpr2 1195 | . . . . 5
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝑋) | 
| 39 |  | simpr3 1196 | . . . . 5
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ 𝐶) | 
| 40 | 38, 39 | eldifd 3961 | . . . 4
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝑋 ∖ 𝐶)) | 
| 41 |  | regsep 23343 | . . . 4
⊢ ((𝐽 ∈ Reg ∧ (𝑋 ∖ 𝐶) ∈ 𝐽 ∧ 𝐴 ∈ (𝑋 ∖ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶))) | 
| 42 | 34, 37, 40, 41 | syl3anc 1372 | . . 3
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶))) | 
| 43 | 33, 42 | reximddv 3170 | . 2
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | 
| 44 |  | rexcom 3289 | . 2
⊢
(∃𝑦 ∈
𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | 
| 45 | 43, 44 | sylib 218 | 1
⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) |