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Mirrors > Home > ILE Home > Th. List > opnssneib | Unicode version |
Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.) |
Ref | Expression |
---|---|
neips.1 |
Ref | Expression |
---|---|
opnssneib |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 520 | . . . . . 6 | |
2 | sseq2 3161 | . . . . . . . . . 10 | |
3 | sseq1 3160 | . . . . . . . . . 10 | |
4 | 2, 3 | anbi12d 465 | . . . . . . . . 9 |
5 | ssid 3157 | . . . . . . . . . 10 | |
6 | 5 | biantrur 301 | . . . . . . . . 9 |
7 | 4, 6 | bitr4di 197 | . . . . . . . 8 |
8 | 7 | rspcev 2825 | . . . . . . 7 |
9 | 8 | adantlr 469 | . . . . . 6 |
10 | 1, 9 | jca 304 | . . . . 5 |
11 | 10 | ex 114 | . . . 4 |
12 | 11 | 3adant1 1004 | . . 3 |
13 | neips.1 | . . . . . 6 | |
14 | 13 | eltopss 12548 | . . . . 5 |
15 | 13 | isnei 12685 | . . . . 5 |
16 | 14, 15 | syldan 280 | . . . 4 |
17 | 16 | 3adant3 1006 | . . 3 |
18 | 12, 17 | sylibrd 168 | . 2 |
19 | ssnei 12692 | . . . 4 | |
20 | 19 | ex 114 | . . 3 |
21 | 20 | 3ad2ant1 1007 | . 2 |
22 | 18, 21 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wrex 2443 wss 3111 cuni 3783 cfv 5182 ctop 12536 cnei 12679 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-top 12537 df-nei 12680 |
This theorem is referenced by: neissex 12706 |
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