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Mirrors > Home > MPE Home > Th. List > opnneiid | Structured version Visualization version GIF version |
Description: Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.) |
Ref | Expression |
---|---|
opnneiid | ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neii2 21241 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑁)) → ∃𝑥 ∈ 𝐽 (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁)) | |
2 | eqss 3813 | . . . . . 6 ⊢ (𝑁 = 𝑥 ↔ (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁)) | |
3 | eleq1a 2873 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 → (𝑁 = 𝑥 → 𝑁 ∈ 𝐽)) | |
4 | 2, 3 | syl5bir 235 | . . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ 𝐽)) |
5 | 4 | rexlimiv 3208 | . . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ 𝐽) |
6 | 1, 5 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑁)) → 𝑁 ∈ 𝐽) |
7 | 6 | ex 402 | . 2 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) → 𝑁 ∈ 𝐽)) |
8 | ssid 3819 | . . 3 ⊢ 𝑁 ⊆ 𝑁 | |
9 | opnneiss 21251 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑁)) | |
10 | 9 | 3exp 1149 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝐽 → (𝑁 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑁)))) |
11 | 8, 10 | mpii 46 | . 2 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝐽 → 𝑁 ∈ ((nei‘𝐽)‘𝑁))) |
12 | 7, 11 | impbid 204 | 1 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ⊆ wss 3769 ‘cfv 6101 Topctop 21026 neicnei 21230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-top 21027 df-nei 21231 |
This theorem is referenced by: 0nei 21261 |
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