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Mirrors > Home > MPE Home > Th. List > opnneissb | Structured version Visualization version GIF version |
Description: An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.) |
Ref | Expression |
---|---|
neips.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
opnneissb | ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neips.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | eltopss 21199 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) → 𝑁 ⊆ 𝑋) |
3 | 2 | adantr 481 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑁)) → 𝑁 ⊆ 𝑋) |
4 | ssid 3910 | . . . . . . 7 ⊢ 𝑁 ⊆ 𝑁 | |
5 | sseq2 3914 | . . . . . . . . 9 ⊢ (𝑔 = 𝑁 → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ 𝑁)) | |
6 | sseq1 3913 | . . . . . . . . 9 ⊢ (𝑔 = 𝑁 → (𝑔 ⊆ 𝑁 ↔ 𝑁 ⊆ 𝑁)) | |
7 | 5, 6 | anbi12d 630 | . . . . . . . 8 ⊢ (𝑔 = 𝑁 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ (𝑆 ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁))) |
8 | 7 | rspcev 3559 | . . . . . . 7 ⊢ ((𝑁 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) |
9 | 4, 8 | mpanr2 700 | . . . . . 6 ⊢ ((𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) |
10 | 9 | ad2ant2l 742 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑁)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) |
11 | 1 | isnei 21395 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
12 | 11 | ad2ant2r 743 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑁)) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
13 | 3, 10, 12 | mpbir2and 709 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) |
14 | 13 | exp43 437 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝐽 → (𝑆 ⊆ 𝑋 → (𝑆 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑆))))) |
15 | 14 | 3imp 1104 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
16 | ssnei 21402 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁) | |
17 | 16 | ex 413 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁)) |
18 | 17 | 3ad2ant1 1126 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁)) |
19 | 15, 18 | impbid 213 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 ⊆ wss 3859 ∪ cuni 4745 ‘cfv 6225 Topctop 21185 neicnei 21389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-top 21186 df-nei 21390 |
This theorem is referenced by: opnneiss 21410 |
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