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Mirrors > Home > ILE Home > Th. List > fiinopn | Unicode version |
Description: The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
Ref | Expression |
---|---|
fiinopn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwg 3441 |
. . . . . . 7
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2 | sseq1 3048 |
. . . . . . . . . . . . . 14
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3 | neeq1 2269 |
. . . . . . . . . . . . . 14
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4 | eleq1 2151 |
. . . . . . . . . . . . . 14
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5 | 2, 3, 4 | 3anbi123d 1249 |
. . . . . . . . . . . . 13
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6 | inteq 3697 |
. . . . . . . . . . . . . . 15
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7 | 6 | eleq1d 2157 |
. . . . . . . . . . . . . 14
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8 | 7 | imbi2d 229 |
. . . . . . . . . . . . 13
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9 | 5, 8 | imbi12d 233 |
. . . . . . . . . . . 12
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10 | sp 1447 |
. . . . . . . . . . . . . 14
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11 | 10 | adantl 272 |
. . . . . . . . . . . . 13
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12 | istopfin 11760 |
. . . . . . . . . . . . 13
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13 | 11, 12 | syl11 31 |
. . . . . . . . . . . 12
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14 | 9, 13 | vtoclg 2680 |
. . . . . . . . . . 11
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15 | 14 | com12 30 |
. . . . . . . . . 10
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16 | 15 | 3exp 1143 |
. . . . . . . . 9
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17 | 16 | com3r 79 |
. . . . . . . 8
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18 | 17 | com4r 86 |
. . . . . . 7
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19 | 1, 18 | syl6bir 163 |
. . . . . 6
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20 | 19 | pm2.43a 51 |
. . . . 5
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21 | 20 | com4l 84 |
. . . 4
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22 | 21 | pm2.43i 49 |
. . 3
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23 | 22 | 3imp 1138 |
. 2
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24 | 23 | com12 30 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-er 6306 df-en 6512 df-fin 6514 df-top 11758 |
This theorem is referenced by: (None) |
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