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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 21726 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ {𝑃} ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) | |
3 | 1, 2 | syl3an3 1161 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Topctop 21501 neicnei 21705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-top 21502 df-nei 21706 |
This theorem is referenced by: opnnei 21728 neindisj2 21731 iscnp4 21871 cnpnei 21872 hausnei2 21961 llynlly 22085 nllyrest 22094 nllyidm 22097 hausllycmp 22102 cldllycmp 22103 txnlly 22245 flimfil 22577 flimopn 22583 fbflim2 22585 hausflimlem 22587 flimcf 22590 flimsncls 22594 fclsnei 22627 fcfnei 22643 cnextcn 22675 utopreg 22861 blnei 23112 cnllycmp 23560 flimcfil 23917 limcflf 24479 rrhre 31262 cvmlift2lem12 32561 |
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