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Mirrors > Home > ILE Home > Th. List > toponsspwpwg | Unicode version |
Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.) |
Ref | Expression |
---|---|
toponsspwpwg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2631 |
. . 3
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2 | rabssab 3109 |
. . . . . 6
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3 | eqcom 2091 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | abbii 2204 |
. . . . . 6
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5 | 2, 4 | sseqtri 3059 |
. . . . 5
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6 | pwpwssunieq 3823 |
. . . . 5
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7 | 5, 6 | sstri 3035 |
. . . 4
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8 | pwexg 4021 |
. . . . 5
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9 | 8 | pwexd 4022 |
. . . 4
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10 | ssexg 3984 |
. . . 4
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11 | 7, 9, 10 | sylancr 406 |
. . 3
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12 | eqeq1 2095 |
. . . . 5
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13 | 12 | rabbidv 2609 |
. . . 4
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14 | df-topon 11764 |
. . . 4
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15 | 13, 14 | fvmptg 5393 |
. . 3
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16 | 1, 11, 15 | syl2anc 404 |
. 2
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17 | 16, 7 | syl6eqss 3077 |
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-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-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 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-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-topon 11764 |
This theorem is referenced by: (None) |
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