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Theorem toponsspwpwg 11774
 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.)
Assertion
Ref Expression
toponsspwpwg TopOn

Proof of Theorem toponsspwpwg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2631 . . 3
2 rabssab 3109 . . . . . 6
3 eqcom 2091 . . . . . . 7
43abbii 2204 . . . . . 6
52, 4sseqtri 3059 . . . . 5
6 pwpwssunieq 3823 . . . . 5
75, 6sstri 3035 . . . 4
8 pwexg 4021 . . . . 5
98pwexd 4022 . . . 4
10 ssexg 3984 . . . 4
117, 9, 10sylancr 406 . . 3
12 eqeq1 2095 . . . . 5
1312rabbidv 2609 . . . 4
14 df-topon 11764 . . . 4 TopOn
1513, 14fvmptg 5393 . . 3 TopOn
161, 11, 15syl2anc 404 . 2 TopOn
1716, 7syl6eqss 3077 1 TopOn
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1290   wcel 1439  cab 2075  crab 2364  cvv 2620   wss 3000  cpw 3433  cuni 3659  cfv 5028  ctop 11750  TopOnctopon 11763 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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