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| Mirrors > Home > MPE Home > Th. List > 0ntr | Structured version Visualization version GIF version | ||
| Description: A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.) |
| Ref | Expression |
|---|---|
| clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| 0ntr | ⊢ (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssdif0 4366 | . . . . 5 ⊢ (𝑋 ⊆ 𝑆 ↔ (𝑋 ∖ 𝑆) = ∅) | |
| 2 | eqss 3999 | . . . . . . . . 9 ⊢ (𝑆 = 𝑋 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑆)) | |
| 3 | fveq2 6906 | . . . . . . . . . . . . 13 ⊢ (𝑆 = 𝑋 → ((int‘𝐽)‘𝑆) = ((int‘𝐽)‘𝑋)) | |
| 4 | clscld.1 | . . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽 | |
| 5 | 4 | ntrtop 23078 | . . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋) |
| 6 | 3, 5 | sylan9eqr 2799 | . . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → ((int‘𝐽)‘𝑆) = 𝑋) |
| 7 | 6 | eqeq1d 2739 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ 𝑋 = ∅)) |
| 8 | 7 | biimpd 229 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = 𝑋) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)) |
| 9 | 8 | ex 412 | . . . . . . . . 9 ⊢ (𝐽 ∈ Top → (𝑆 = 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))) |
| 10 | 2, 9 | biimtrrid 243 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑆 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑆) → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅))) |
| 11 | 10 | expd 415 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → (𝑋 ⊆ 𝑆 → (((int‘𝐽)‘𝑆) = ∅ → 𝑋 = ∅)))) |
| 12 | 11 | com34 91 | . . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → (((int‘𝐽)‘𝑆) = ∅ → (𝑋 ⊆ 𝑆 → 𝑋 = ∅)))) |
| 13 | 12 | imp32 418 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ⊆ 𝑆 → 𝑋 = ∅)) |
| 14 | 1, 13 | biimtrrid 243 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → ((𝑋 ∖ 𝑆) = ∅ → 𝑋 = ∅)) |
| 15 | 14 | necon3d 2961 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ≠ ∅ → (𝑋 ∖ 𝑆) ≠ ∅)) |
| 16 | 15 | imp 406 | . 2 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) ∧ 𝑋 ≠ ∅) → (𝑋 ∖ 𝑆) ≠ ∅) |
| 17 | 16 | an32s 652 | 1 ⊢ (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 intcnt 23025 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-ntr 23028 |
| This theorem is referenced by: (None) |
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