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Mirrors > Home > ILE Home > Th. List > uniopn | Unicode version |
Description: The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
uniopn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 13469 |
. . . . 5
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2 | 1 | ibi 176 |
. . . 4
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3 | 2 | simpld 112 |
. . 3
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4 | elpw2g 4156 |
. . . . . . . 8
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5 | 4 | biimpar 297 |
. . . . . . 7
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6 | sseq1 3178 |
. . . . . . . . 9
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7 | unieq 3818 |
. . . . . . . . . 10
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8 | 7 | eleq1d 2246 |
. . . . . . . . 9
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9 | 6, 8 | imbi12d 234 |
. . . . . . . 8
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10 | 9 | spcgv 2824 |
. . . . . . 7
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11 | 5, 10 | syl 14 |
. . . . . 6
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12 | 11 | com23 78 |
. . . . 5
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13 | 12 | ex 115 |
. . . 4
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14 | 13 | pm2.43d 50 |
. . 3
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15 | 3, 14 | mpid 42 |
. 2
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16 | 15 | imp 124 |
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-ext 2159 ax-sep 4121 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-uni 3810 df-top 13468 |
This theorem is referenced by: iunopn 13472 unopn 13475 0opn 13476 topopn 13478 tgtop 13538 ntropn 13587 neipsm 13624 unimopn 13956 metrest 13976 cnopncntop 14007 |
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