![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4810 | . 2 ⊢ (𝑃 ∈ 𝑁 → {𝑃} ⊆ 𝑁) | |
2 | opnneiss 22613 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ {𝑃} ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) | |
3 | 1, 2 | syl3an3 1165 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 ⊆ wss 3947 {csn 4627 ‘cfv 6540 Topctop 22386 neicnei 22592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-top 22387 df-nei 22593 |
This theorem is referenced by: opnnei 22615 neindisj2 22618 iscnp4 22758 cnpnei 22759 hausnei2 22848 llynlly 22972 nllyrest 22981 nllyidm 22984 hausllycmp 22989 cldllycmp 22990 txnlly 23132 flimfil 23464 flimopn 23470 fbflim2 23472 hausflimlem 23474 flimcf 23477 flimsncls 23481 fclsnei 23514 fcfnei 23530 cnextcn 23562 utopreg 23748 blnei 24002 cnllycmp 24463 flimcfil 24822 limcflf 25389 rrhre 32989 cvmlift2lem12 34293 |
Copyright terms: Public domain | W3C validator |