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Mirrors > Home > ILE Home > Th. List > istopg | Unicode version |
Description: Express the predicate
" is a
topology". See istopfin 12170 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 3513 | . . . . 5 | |
2 | eleq2 2203 | . . . . 5 | |
3 | 1, 2 | raleqbidv 2638 | . . . 4 |
4 | eleq2 2203 | . . . . . 6 | |
5 | 4 | raleqbi1dv 2634 | . . . . 5 |
6 | 5 | raleqbi1dv 2634 | . . . 4 |
7 | 3, 6 | anbi12d 464 | . . 3 |
8 | df-top 12168 | . . 3 | |
9 | 7, 8 | elab2g 2831 | . 2 |
10 | df-ral 2421 | . . . 4 | |
11 | elpw2g 4081 | . . . . . 6 | |
12 | 11 | imbi1d 230 | . . . . 5 |
13 | 12 | albidv 1796 | . . . 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 1329 wceq 1331 wcel 1480 wral 2416 cin 3070 wss 3071 cpw 3510 cuni 3736 ctop 12167 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 df-top 12168 |
This theorem is referenced by: istopfin 12170 uniopn 12171 inopn 12173 tgcl 12236 distop 12257 epttop 12262 |
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