Proof of Theorem ist1-2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | topontop 22919 | . . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 2 |  | eqid 2737 | . . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 3 | 2 | ist1 23329 | . . . 4
⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽))) | 
| 4 | 3 | baib 535 | . . 3
⊢ (𝐽 ∈ Top → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽))) | 
| 5 | 1, 4 | syl 17 | . 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽))) | 
| 6 |  | toponuni 22920 | . . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 7 | 6 | raleqdv 3326 | . 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽))) | 
| 8 | 1 | adantr 480 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top) | 
| 9 |  | eltop2 22982 | . . . . . 6
⊢ (𝐽 ∈ Top → ((∪ 𝐽
∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽
∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 10 | 8, 9 | syl 17 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((∪
𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 11 | 6 | eleq2d 2827 | . . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐽)) | 
| 12 | 11 | biimpa 476 | . . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝐽) | 
| 13 | 12 | snssd 4809 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ ∪ 𝐽) | 
| 14 | 2 | iscld2 23036 | . . . . . 6
⊢ ((𝐽 ∈ Top ∧ {𝑦} ⊆ ∪ 𝐽)
→ ({𝑦} ∈
(Clsd‘𝐽) ↔
(∪ 𝐽 ∖ {𝑦}) ∈ 𝐽)) | 
| 15 | 8, 13, 14 | syl2anc 584 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽)) | 
| 16 | 6 | adantr 480 | . . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑋 = ∪ 𝐽) | 
| 17 | 16 | eleq2d 2827 | . . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽)) | 
| 18 | 17 | imbi1d 341 | . . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))) | 
| 19 |  | con1b 358 | . . . . . . . . 9
⊢ ((¬
𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦)) | 
| 20 |  | df-ne 2941 | . . . . . . . . . 10
⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | 
| 21 | 20 | imbi1i 349 | . . . . . . . . 9
⊢ ((𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 22 |  | disjsn 4711 | . . . . . . . . . . . . . . 15
⊢ ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑜) | 
| 23 |  | elssuni 4937 | . . . . . . . . . . . . . . . 16
⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽) | 
| 24 |  | reldisj 4453 | . . . . . . . . . . . . . . . 16
⊢ (𝑜 ⊆ ∪ 𝐽
→ ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽
∖ {𝑦}))) | 
| 25 | 23, 24 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝑜 ∈ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) | 
| 26 | 22, 25 | bitr3id 285 | . . . . . . . . . . . . . 14
⊢ (𝑜 ∈ 𝐽 → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) | 
| 27 | 26 | anbi2d 630 | . . . . . . . . . . . . 13
⊢ (𝑜 ∈ 𝐽 → ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 28 | 27 | rexbiia 3092 | . . . . . . . . . . . 12
⊢
(∃𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) | 
| 29 |  | rexanali 3102 | . . . . . . . . . . . 12
⊢
(∃𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜)) | 
| 30 | 28, 29 | bitr3i 277 | . . . . . . . . . . 11
⊢
(∃𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜)) | 
| 31 | 30 | con2bii 357 | . . . . . . . . . 10
⊢
(∀𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) | 
| 32 | 31 | imbi1i 349 | . . . . . . . . 9
⊢
((∀𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦)) | 
| 33 | 19, 21, 32 | 3bitr4ri 304 | . . . . . . . 8
⊢
((∀𝑜 ∈
𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 34 | 33 | imbi2i 336 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))) | 
| 35 |  | eldifsn 4786 | . . . . . . . . 9
⊢ (𝑥 ∈ (∪ 𝐽
∖ {𝑦}) ↔ (𝑥 ∈ ∪ 𝐽
∧ 𝑥 ≠ 𝑦)) | 
| 36 | 35 | imbi1i 349 | . . . . . . . 8
⊢ ((𝑥 ∈ (∪ 𝐽
∖ {𝑦}) →
∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 37 |  | impexp 450 | . . . . . . . 8
⊢ (((𝑥 ∈ ∪ 𝐽
∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))) | 
| 38 | 36, 37 | bitri 275 | . . . . . . 7
⊢ ((𝑥 ∈ (∪ 𝐽
∖ {𝑦}) →
∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))) | 
| 39 | 18, 34, 38 | 3bitr4g 314 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))) | 
| 40 | 39 | ralbidv2 3174 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) | 
| 41 | 10, 15, 40 | 3bitr4d 311 | . . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | 
| 42 | 41 | ralbidva 3176 | . . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | 
| 43 |  | ralcom 3289 | . . 3
⊢
(∀𝑦 ∈
𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) | 
| 44 | 42, 43 | bitrdi 287 | . 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | 
| 45 | 5, 7, 44 | 3bitr2d 307 | 1
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |