Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > distop | Unicode version |
Description: The discrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
distop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniss 3757 | . . . . . 6 | |
2 | unipw 4139 | . . . . . 6 | |
3 | 1, 2 | sseqtrdi 3145 | . . . . 5 |
4 | vuniex 4360 | . . . . . 6 | |
5 | 4 | elpw 3516 | . . . . 5 |
6 | 3, 5 | sylibr 133 | . . . 4 |
7 | 6 | ax-gen 1425 | . . 3 |
8 | 7 | a1i 9 | . 2 |
9 | velpw 3517 | . . . . . 6 | |
10 | velpw 3517 | . . . . . . . 8 | |
11 | ssinss1 3305 | . . . . . . . . . 10 | |
12 | 11 | a1i 9 | . . . . . . . . 9 |
13 | vex 2689 | . . . . . . . . . . 11 | |
14 | 13 | inex2 4063 | . . . . . . . . . 10 |
15 | 14 | elpw 3516 | . . . . . . . . 9 |
16 | 12, 15 | syl6ibr 161 | . . . . . . . 8 |
17 | 10, 16 | sylbi 120 | . . . . . . 7 |
18 | 17 | com12 30 | . . . . . 6 |
19 | 9, 18 | sylbi 120 | . . . . 5 |
20 | 19 | ralrimiv 2504 | . . . 4 |
21 | 20 | rgen 2485 | . . 3 |
22 | 21 | a1i 9 | . 2 |
23 | pwexg 4104 | . . 3 | |
24 | istopg 12166 | . . 3 | |
25 | 23, 24 | syl 14 | . 2 |
26 | 8, 22, 25 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wcel 1480 wral 2416 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 ctop 12164 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-un 4355 |
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-rex 2422 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-uni 3737 df-top 12165 |
This theorem is referenced by: topnex 12255 distopon 12256 distps 12260 discld 12305 restdis 12353 txdis 12446 |
Copyright terms: Public domain | W3C validator |