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| Mirrors > Home > MPE Home > Th. List > opnneip | Structured version Visualization version GIF version | ||
| Description: An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.) |
| Ref | Expression |
|---|---|
| opnneip | ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssi 4808 | . 2 ⊢ (𝑃 ∈ 𝑁 → {𝑃} ⊆ 𝑁) | |
| 2 | opnneiss 23126 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ {𝑃} ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) | |
| 3 | 1, 2 | syl3an3 1166 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 ‘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: opnnei 23128 neindisj2 23131 iscnp4 23271 cnpnei 23272 hausnei2 23361 llynlly 23485 nllyrest 23494 nllyidm 23497 hausllycmp 23502 cldllycmp 23503 txnlly 23645 flimfil 23977 flimopn 23983 fbflim2 23985 hausflimlem 23987 flimcf 23990 flimsncls 23994 fclsnei 24027 fcfnei 24043 cnextcn 24075 utopreg 24261 blnei 24515 cnllycmp 24988 flimcfil 25348 limcflf 25916 rrhre 34022 cvmlift2lem12 35319 |
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