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Mirrors > Home > MPE Home > Th. List > istopg | Structured version Visualization version Unicode version |
Description: Express the predicate
" is a
topology." See istop2g 20701 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 4161 | . . . . 5 | |
2 | eleq2 2690 | . . . . 5 | |
3 | 1, 2 | raleqbidv 3152 | . . . 4 |
4 | eleq2 2690 | . . . . . 6 | |
5 | 4 | raleqbi1dv 3146 | . . . . 5 |
6 | 5 | raleqbi1dv 3146 | . . . 4 |
7 | 3, 6 | anbi12d 747 | . . 3 |
8 | df-top 20699 | . . 3 | |
9 | 7, 8 | elab2g 3353 | . 2 |
10 | df-ral 2917 | . . . 4 | |
11 | elpw2g 4827 | . . . . . 6 | |
12 | 11 | imbi1d 331 | . . . . 5 |
13 | 12 | albidv 1849 | . . . 4 |
14 | 10, 13 | syl5bb 272 | . . 3 |
15 | 14 | anbi1d 741 | . 2 |
16 | 9, 15 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wcel 1990 wral 2912 cin 3573 wss 3574 cpw 4158 cuni 4436 ctop 20698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-in 3581 df-ss 3588 df-pw 4160 df-top 20699 |
This theorem is referenced by: istop2g 20701 uniopn 20702 inopn 20704 tgcl 20773 distop 20799 indistopon 20805 fctop 20808 cctop 20810 ppttop 20811 epttop 20813 mretopd 20896 toponmre 20897 neiptoptop 20935 kgentopon 21341 qtoptop2 21502 filconn 21687 utoptop 22038 neibastop1 32354 |
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