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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 4812 | . 2 ⊢ (𝑃 ∈ 𝑁 → {𝑃} ⊆ 𝑁) | |
2 | opnneiss 23141 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ {𝑃} ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) | |
3 | 1, 2 | syl3an3 1164 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2105 ⊆ wss 3962 {csn 4630 ‘cfv 6562 Topctop 22914 neicnei 23120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-top 22915 df-nei 23121 |
This theorem is referenced by: opnnei 23143 neindisj2 23146 iscnp4 23286 cnpnei 23287 hausnei2 23376 llynlly 23500 nllyrest 23509 nllyidm 23512 hausllycmp 23517 cldllycmp 23518 txnlly 23660 flimfil 23992 flimopn 23998 fbflim2 24000 hausflimlem 24002 flimcf 24005 flimsncls 24009 fclsnei 24042 fcfnei 24058 cnextcn 24090 utopreg 24276 blnei 24530 cnllycmp 25001 flimcfil 25361 limcflf 25930 rrhre 33983 cvmlift2lem12 35298 |
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