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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 4741 | . 2 ⊢ (𝑃 ∈ 𝑁 → {𝑃} ⊆ 𝑁) | |
2 | opnneiss 22269 | . 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 2106 ⊆ wss 3887 {csn 4561 ‘cfv 6433 Topctop 22042 neicnei 22248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-top 22043 df-nei 22249 |
This theorem is referenced by: opnnei 22271 neindisj2 22274 iscnp4 22414 cnpnei 22415 hausnei2 22504 llynlly 22628 nllyrest 22637 nllyidm 22640 hausllycmp 22645 cldllycmp 22646 txnlly 22788 flimfil 23120 flimopn 23126 fbflim2 23128 hausflimlem 23130 flimcf 23133 flimsncls 23137 fclsnei 23170 fcfnei 23186 cnextcn 23218 utopreg 23404 blnei 23658 cnllycmp 24119 flimcfil 24478 limcflf 25045 rrhre 31971 cvmlift2lem12 33276 |
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