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Mirrors > Home > ILE Home > Th. List > istopg | Unicode version |
Description: Express the predicate
" is a
topology". See istopfin 12398 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 3546 | . . . . 5 | |
2 | eleq2 2221 | . . . . 5 | |
3 | 1, 2 | raleqbidv 2664 | . . . 4 |
4 | eleq2 2221 | . . . . . 6 | |
5 | 4 | raleqbi1dv 2660 | . . . . 5 |
6 | 5 | raleqbi1dv 2660 | . . . 4 |
7 | 3, 6 | anbi12d 465 | . . 3 |
8 | df-top 12396 | . . 3 | |
9 | 7, 8 | elab2g 2859 | . 2 |
10 | df-ral 2440 | . . . 4 | |
11 | elpw2g 4117 | . . . . . 6 | |
12 | 11 | imbi1d 230 | . . . . 5 |
13 | 12 | albidv 1804 | . . . 4 |
14 | 10, 13 | syl5bb 191 | . . 3 |
15 | 14 | anbi1d 461 | . 2 |
16 | 9, 15 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wcel 2128 wral 2435 cin 3101 wss 3102 cpw 3543 cuni 3772 ctop 12395 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-sep 4082 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-in 3108 df-ss 3115 df-pw 3545 df-top 12396 |
This theorem is referenced by: istopfin 12398 uniopn 12399 inopn 12401 tgcl 12464 distop 12485 epttop 12490 |
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