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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnbnd | Structured version Visualization version Unicode version |
Description: A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.) |
Ref | Expression |
---|---|
opnbnd.1 |
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Ref | Expression |
---|---|
opnbnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdif 3841 |
. . . . 5
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2 | 1 | a1i 11 |
. . . 4
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3 | ineq1 3629 |
. . . . 5
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4 | 3 | eqeq1d 2455 |
. . . 4
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5 | 2, 4 | syl5ibcom 224 |
. . 3
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6 | opnbnd.1 |
. . . . . . 7
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7 | 6 | ntrss2 20084 |
. . . . . 6
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8 | 7 | adantr 467 |
. . . . 5
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9 | inssdif0 3836 |
. . . . . 6
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10 | 6 | sscls 20083 |
. . . . . . . . . 10
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11 | df-ss 3420 |
. . . . . . . . . 10
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12 | 10, 11 | sylib 200 |
. . . . . . . . 9
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13 | 12 | eqcomd 2459 |
. . . . . . . 8
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14 | eqimss 3486 |
. . . . . . . 8
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15 | 13, 14 | syl 17 |
. . . . . . 7
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16 | sstr 3442 |
. . . . . . 7
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17 | 15, 16 | sylan 474 |
. . . . . 6
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18 | 9, 17 | sylan2br 479 |
. . . . 5
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19 | 8, 18 | eqssd 3451 |
. . . 4
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20 | 19 | ex 436 |
. . 3
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21 | 5, 20 | impbid 194 |
. 2
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22 | 6 | isopn3 20094 |
. 2
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23 | 6 | topbnd 30992 |
. . . 4
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24 | 23 | ineq2d 3636 |
. . 3
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25 | 24 | eqeq1d 2455 |
. 2
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26 | 21, 22, 25 | 3bitr4d 289 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-top 19933 df-cld 20046 df-ntr 20047 df-cls 20048 |
This theorem is referenced by: cldbnd 30994 |
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