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Mirrors > Home > MPE Home > Th. List > opnneiss | Structured version Visualization version GIF version |
Description: An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
Ref | Expression |
---|---|
opnneiss | ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1134 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ 𝑁) | |
2 | eqid 2821 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | eltopss 21509 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) → 𝑁 ⊆ ∪ 𝐽) |
4 | sstr 3975 | . . . . 5 ⊢ ((𝑆 ⊆ 𝑁 ∧ 𝑁 ⊆ ∪ 𝐽) → 𝑆 ⊆ ∪ 𝐽) | |
5 | 4 | ancoms 461 | . . . 4 ⊢ ((𝑁 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ ∪ 𝐽) |
6 | 3, 5 | stoic3 1773 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ ∪ 𝐽) |
7 | 2 | opnneissb 21716 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
8 | 6, 7 | syld3an3 1405 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
9 | 1, 8 | mpbid 234 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 ⊆ wss 3936 ∪ cuni 4832 ‘cfv 6350 Topctop 21495 neicnei 21699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-top 21496 df-nei 21700 |
This theorem is referenced by: opnneip 21721 tpnei 21723 topssnei 21726 opnneiid 21728 neissex 21729 cmpkgen 22153 |
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