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Mirrors > Home > MPE Home > Th. List > uniopn | Structured version Visualization version Unicode version |
Description: The union of a subset 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 19937 |
. . . . 5
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2 | 1 | ibi 245 |
. . . 4
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3 | 2 | simpld 461 |
. . 3
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4 | elpw2g 4569 |
. . . . . . . 8
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5 | 4 | biimpar 488 |
. . . . . . 7
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6 | sseq1 3455 |
. . . . . . . . 9
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7 | unieq 4209 |
. . . . . . . . . 10
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8 | 7 | eleq1d 2515 |
. . . . . . . . 9
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9 | 6, 8 | imbi12d 322 |
. . . . . . . 8
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10 | 9 | spcgv 3136 |
. . . . . . 7
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11 | 5, 10 | syl 17 |
. . . . . 6
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12 | 11 | com23 81 |
. . . . 5
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13 | 12 | ex 436 |
. . . 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 431 |
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-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 |
This theorem depends on definitions: df-bi 189 df-an 373 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ral 2744 df-rex 2745 df-v 3049 df-in 3413 df-ss 3420 df-pw 3955 df-uni 4202 df-top 19933 |
This theorem is referenced by: iunopn 19940 unopn 19945 0opn 19946 topopn 19948 tgtop 20001 ntropn 20076 toponmre 20121 neips 20141 txcmplem1 20668 unimopn 21523 metrest 21551 locfinreflem 28679 cvmscld 30008 mblfinlem3 31991 mblfinlem4 31992 ismblfin 31993 cnopn 37385 |
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