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Mirrors > Home > MPE Home > Th. List > ntrval | Structured version Unicode version |
Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 |
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Ref | Expression |
---|---|
ntrval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 |
. . . . 5
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2 | 1 | ntrfval 18770 |
. . . 4
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3 | 2 | fveq1d 5804 |
. . 3
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4 | 3 | adantr 465 |
. 2
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5 | 1 | topopn 18661 |
. . . . 5
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6 | elpw2g 4566 |
. . . . 5
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7 | 5, 6 | syl 16 |
. . . 4
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8 | 7 | biimpar 485 |
. . 3
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9 | inex1g 4546 |
. . . . 5
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10 | 9 | adantr 465 |
. . . 4
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11 | uniexg 6490 |
. . . 4
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12 | 10, 11 | syl 16 |
. . 3
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13 | pweq 3974 |
. . . . . 6
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14 | 13 | ineq2d 3663 |
. . . . 5
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15 | 14 | unieqd 4212 |
. . . 4
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16 | eqid 2454 |
. . . 4
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17 | 15, 16 | fvmptg 5884 |
. . 3
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18 | 8, 12, 17 | syl2anc 661 |
. 2
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19 | 4, 18 | eqtrd 2495 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-top 18645 df-ntr 18766 |
This theorem is referenced by: ntropn 18795 clsval2 18796 ntrss2 18803 ssntr 18804 isopn3 18812 ntreq0 18823 |
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