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Mirrors > Home > ILE Home > Th. List > istopg | Unicode version |
Description: Express the predicate
" is a
topology". See istopfin 12094 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 3483 | . . . . 5 | |
2 | eleq2 2181 | . . . . 5 | |
3 | 1, 2 | raleqbidv 2615 | . . . 4 |
4 | eleq2 2181 | . . . . . 6 | |
5 | 4 | raleqbi1dv 2611 | . . . . 5 |
6 | 5 | raleqbi1dv 2611 | . . . 4 |
7 | 3, 6 | anbi12d 464 | . . 3 |
8 | df-top 12092 | . . 3 | |
9 | 7, 8 | elab2g 2804 | . 2 |
10 | df-ral 2398 | . . . 4 | |
11 | elpw2g 4051 | . . . . . 6 | |
12 | 11 | imbi1d 230 | . . . . 5 |
13 | 12 | albidv 1780 | . . . 4 |
14 | 10, 13 | syl5bb 191 | . . 3 |
15 | 14 | anbi1d 460 | . 2 |
16 | 9, 15 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 cpw 3480 cuni 3706 ctop 12091 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-top 12092 |
This theorem is referenced by: istopfin 12094 uniopn 12095 inopn 12097 tgcl 12160 distop 12181 epttop 12186 |
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