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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 4744 | . 2 ⊢ (𝑃 ∈ 𝑁 → {𝑃} ⊆ 𝑁) | |
2 | opnneiss 21729 | . 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 2113 ⊆ wss 3939 {csn 4570 ‘cfv 6358 Topctop 21504 neicnei 21708 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-top 21505 df-nei 21709 |
This theorem is referenced by: opnnei 21731 neindisj2 21734 iscnp4 21874 cnpnei 21875 hausnei2 21964 llynlly 22088 nllyrest 22097 nllyidm 22100 hausllycmp 22105 cldllycmp 22106 txnlly 22248 flimfil 22580 flimopn 22586 fbflim2 22588 hausflimlem 22590 flimcf 22593 flimsncls 22597 fclsnei 22630 fcfnei 22646 cnextcn 22678 utopreg 22864 blnei 23115 cnllycmp 23563 flimcfil 23920 limcflf 24482 rrhre 31266 cvmlift2lem12 32565 |
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