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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 |
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Ref | Expression |
---|---|
opnssneib |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 528 |
. . . . . 6
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2 | sseq2 3179 |
. . . . . . . . . 10
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3 | sseq1 3178 |
. . . . . . . . . 10
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4 | 2, 3 | anbi12d 473 |
. . . . . . . . 9
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5 | ssid 3175 |
. . . . . . . . . 10
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6 | 5 | biantrur 303 |
. . . . . . . . 9
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7 | 4, 6 | bitr4di 198 |
. . . . . . . 8
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8 | 7 | rspcev 2841 |
. . . . . . 7
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9 | 8 | adantlr 477 |
. . . . . 6
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10 | 1, 9 | jca 306 |
. . . . 5
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11 | 10 | ex 115 |
. . . 4
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12 | 11 | 3adant1 1015 |
. . 3
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13 | neips.1 |
. . . . . 6
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14 | 13 | eltopss 13140 |
. . . . 5
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15 | 13 | isnei 13277 |
. . . . 5
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16 | 14, 15 | syldan 282 |
. . . 4
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17 | 16 | 3adant3 1017 |
. . 3
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18 | 12, 17 | sylibrd 169 |
. 2
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19 | ssnei 13284 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | ex 115 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | 3ad2ant1 1018 |
. 2
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22 | 18, 21 | impbid 129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-top 13129 df-nei 13272 |
This theorem is referenced by: neissex 13298 |
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