| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > opnssneib | Structured version Visualization version GIF version | ||
| Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.) |
| Ref | Expression |
|---|---|
| neips.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| opnssneib | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simplr 769 | . . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → 𝑁 ⊆ 𝑋) | |
| 2 | sseq2 4010 | . . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ 𝑆)) | |
| 3 | sseq1 4009 | . . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑔 ⊆ 𝑁 ↔ 𝑆 ⊆ 𝑁)) | |
| 4 | 2, 3 | anbi12d 632 | . . . . . . . . 9 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁))) |
| 5 | ssid 4006 | . . . . . . . . . 10 ⊢ 𝑆 ⊆ 𝑆 | |
| 6 | 5 | biantrur 530 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝑁 ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁)) |
| 7 | 4, 6 | bitr4di 289 | . . . . . . . 8 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ 𝑆 ⊆ 𝑁)) |
| 8 | 7 | rspcev 3622 | . . . . . . 7 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) |
| 9 | 8 | adantlr 715 | . . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) |
| 10 | 1, 9 | jca 511 | . . . . 5 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))) |
| 11 | 10 | ex 412 | . . . 4 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
| 12 | 11 | 3adant1 1131 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
| 13 | neips.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
| 14 | 13 | eltopss 22913 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋) |
| 15 | 13 | isnei 23111 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
| 16 | 14, 15 | syldan 591 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
| 17 | 16 | 3adant3 1133 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
| 18 | 12, 17 | sylibrd 259 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
| 19 | ssnei 23118 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁) | |
| 20 | 19 | ex 412 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁)) |
| 21 | 20 | 3ad2ant1 1134 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁)) |
| 22 | 18, 21 | impbid 212 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 neicnei 23105 |
| 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 |
| 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-nei 23106 |
| This theorem is referenced by: neissex 23135 |
| Copyright terms: Public domain | W3C validator |