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Mirrors > Home > ILE Home > Th. List > opnneiss | GIF version |
Description: An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
Ref | Expression |
---|---|
opnneiss | ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 994 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ 𝑁) | |
2 | eqid 2170 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | eltopss 12760 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) → 𝑁 ⊆ ∪ 𝐽) |
4 | sstr 3155 | . . . . 5 ⊢ ((𝑆 ⊆ 𝑁 ∧ 𝑁 ⊆ ∪ 𝐽) → 𝑆 ⊆ ∪ 𝐽) | |
5 | 4 | ancoms 266 | . . . 4 ⊢ ((𝑁 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ ∪ 𝐽) |
6 | 3, 5 | stoic3 1424 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑆 ⊆ ∪ 𝐽) |
7 | 2 | opnneissb 12908 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
8 | 6, 7 | syld3an3 1278 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) |
9 | 1, 8 | mpbid 146 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 973 ∈ wcel 2141 ⊆ wss 3121 ∪ cuni 3794 ‘cfv 5196 Topctop 12748 neicnei 12891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-top 12749 df-nei 12892 |
This theorem is referenced by: opnneip 12912 tpnei 12913 topssnei 12915 opnneiid 12917 neissex 12918 |
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