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Mirrors > Home > MPE Home > Th. List > elnei | Structured version Visualization version GIF version |
Description: A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.) |
Ref | Expression |
---|---|
elnei | ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑃 ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnei 21292 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → {𝑃} ⊆ 𝑁) | |
2 | 1 | 3adant2 1165 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → {𝑃} ⊆ 𝑁) |
3 | snssg 4536 | . . 3 ⊢ (𝑃 ∈ 𝐴 → (𝑃 ∈ 𝑁 ↔ {𝑃} ⊆ 𝑁)) | |
4 | 3 | 3ad2ant2 1168 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑃 ∈ 𝑁 ↔ {𝑃} ⊆ 𝑁)) |
5 | 2, 4 | mpbird 249 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑃 ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1111 ∈ wcel 2164 ⊆ wss 3798 {csn 4399 ‘cfv 6127 Topctop 21075 neicnei 21279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-top 21076 df-nei 21280 |
This theorem is referenced by: gneispa 39267 |
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