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Mirrors > Home > ILE Home > Th. List > istopg | Unicode version |
Description: Express the predicate
" is a
topology". See istopfin 12792 for
another characterization using nonempty finite intersections instead of
binary intersections.
Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
istopg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 3569 | . . . . 5 | |
2 | eleq2 2234 | . . . . 5 | |
3 | 1, 2 | raleqbidv 2677 | . . . 4 |
4 | eleq2 2234 | . . . . . 6 | |
5 | 4 | raleqbi1dv 2673 | . . . . 5 |
6 | 5 | raleqbi1dv 2673 | . . . 4 |
7 | 3, 6 | anbi12d 470 | . . 3 |
8 | df-top 12790 | . . 3 | |
9 | 7, 8 | elab2g 2877 | . 2 |
10 | df-ral 2453 | . . . 4 | |
11 | elpw2g 4142 | . . . . . 6 | |
12 | 11 | imbi1d 230 | . . . . 5 |
13 | 12 | albidv 1817 | . . . 4 |
14 | 10, 13 | syl5bb 191 | . . 3 |
15 | 14 | anbi1d 462 | . 2 |
16 | 9, 15 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wcel 2141 wral 2448 cin 3120 wss 3121 cpw 3566 cuni 3796 ctop 12789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-in 3127 df-ss 3134 df-pw 3568 df-top 12790 |
This theorem is referenced by: istopfin 12792 uniopn 12793 inopn 12795 tgcl 12858 distop 12879 epttop 12884 |
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