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Mirrors > Home > ILE Home > Th. List > fiinbas | Unicode version |
Description: If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
fiinbas |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3190 |
. . . . . . . 8
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2 | eleq2 2253 |
. . . . . . . . . 10
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3 | sseq1 3193 |
. . . . . . . . . 10
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4 | 2, 3 | anbi12d 473 |
. . . . . . . . 9
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5 | 4 | rspcev 2856 |
. . . . . . . 8
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6 | 1, 5 | mpanr2 438 |
. . . . . . 7
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7 | 6 | ralrimiva 2563 |
. . . . . 6
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8 | 7 | a1i 9 |
. . . . 5
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9 | 8 | ralimdv 2558 |
. . . 4
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10 | 9 | ralimdv 2558 |
. . 3
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11 | isbasis2g 13942 |
. . 3
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12 | 10, 11 | sylibrd 169 |
. 2
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13 | 12 | 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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-uni 3825 df-bases 13940 |
This theorem is referenced by: qtopbasss 14418 |
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