Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > opnneiid | GIF version |
Description: Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.) |
Ref | Expression |
---|---|
opnneiid | ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neii2 12318 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑁)) → ∃𝑥 ∈ 𝐽 (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁)) | |
2 | eqss 3112 | . . . . . 6 ⊢ (𝑁 = 𝑥 ↔ (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁)) | |
3 | eleq1a 2211 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 → (𝑁 = 𝑥 → 𝑁 ∈ 𝐽)) | |
4 | 2, 3 | syl5bir 152 | . . . . 5 ⊢ (𝑥 ∈ 𝐽 → ((𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ 𝐽)) |
5 | 4 | rexlimiv 2543 | . . . 4 ⊢ (∃𝑥 ∈ 𝐽 (𝑁 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ 𝐽) |
6 | 1, 5 | syl 14 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑁)) → 𝑁 ∈ 𝐽) |
7 | 6 | ex 114 | . 2 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) → 𝑁 ∈ 𝐽)) |
8 | ssid 3117 | . . 3 ⊢ 𝑁 ⊆ 𝑁 | |
9 | opnneiss 12327 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑁)) | |
10 | 9 | 3exp 1180 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝐽 → (𝑁 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑁)))) |
11 | 8, 10 | mpii 44 | . 2 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝐽 → 𝑁 ∈ ((nei‘𝐽)‘𝑁))) |
12 | 7, 11 | impbid 128 | 1 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 ‘cfv 5123 Topctop 12164 neicnei 12307 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-nei 12308 |
This theorem is referenced by: 0nei 12335 |
Copyright terms: Public domain | W3C validator |