Step | Hyp | Ref
| Expression |
1 | | simpll3 1215 |
. . . . 5
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ ((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄)))) β (π½ βΎt π΄) β Conn) |
2 | | simpll1 1213 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π½ β (TopOnβπ)) |
3 | | simpll2 1214 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π΄ β π) |
4 | | simplrl 776 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π₯ β π½) |
5 | | simplrr 777 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π¦ β π½) |
6 | | simprl1 1219 |
. . . . . . . . 9
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π₯ β© ((clsβπ½)βπ΄)) β β
) |
7 | | n0 4310 |
. . . . . . . . 9
β’ ((π₯ β© ((clsβπ½)βπ΄)) β β
β βπ§ π§ β (π₯ β© ((clsβπ½)βπ΄))) |
8 | 6, 7 | sylib 217 |
. . . . . . . 8
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β βπ§ π§ β (π₯ β© ((clsβπ½)βπ΄))) |
9 | 2 | adantr 482 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π½ β (TopOnβπ)) |
10 | | topontop 22285 |
. . . . . . . . . 10
β’ (π½ β (TopOnβπ) β π½ β Top) |
11 | 9, 10 | syl 17 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π½ β Top) |
12 | 3 | adantr 482 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π΄ β π) |
13 | | toponuni 22286 |
. . . . . . . . . . 11
β’ (π½ β (TopOnβπ) β π = βͺ π½) |
14 | 9, 13 | syl 17 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π = βͺ π½) |
15 | 12, 14 | sseqtrd 3988 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π΄ β βͺ π½) |
16 | | simpr 486 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π§ β (π₯ β© ((clsβπ½)βπ΄))) |
17 | 16 | elin2d 4163 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π§ β ((clsβπ½)βπ΄)) |
18 | 4 | adantr 482 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π₯ β π½) |
19 | 16 | elin1d 4162 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β π§ β π₯) |
20 | | eqid 2733 |
. . . . . . . . . 10
β’ βͺ π½ =
βͺ π½ |
21 | 20 | clsndisj 22449 |
. . . . . . . . 9
β’ (((π½ β Top β§ π΄ β βͺ π½
β§ π§ β
((clsβπ½)βπ΄)) β§ (π₯ β π½ β§ π§ β π₯)) β (π₯ β© π΄) β β
) |
22 | 11, 15, 17, 18, 19, 21 | syl32anc 1379 |
. . . . . . . 8
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π₯ β© ((clsβπ½)βπ΄))) β (π₯ β© π΄) β β
) |
23 | 8, 22 | exlimddv 1939 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π₯ β© π΄) β β
) |
24 | | simprl2 1220 |
. . . . . . . . 9
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π¦ β© ((clsβπ½)βπ΄)) β β
) |
25 | | n0 4310 |
. . . . . . . . 9
β’ ((π¦ β© ((clsβπ½)βπ΄)) β β
β βπ§ π§ β (π¦ β© ((clsβπ½)βπ΄))) |
26 | 24, 25 | sylib 217 |
. . . . . . . 8
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β βπ§ π§ β (π¦ β© ((clsβπ½)βπ΄))) |
27 | 2 | adantr 482 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π½ β (TopOnβπ)) |
28 | 27, 10 | syl 17 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π½ β Top) |
29 | 3 | adantr 482 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π΄ β π) |
30 | 27, 13 | syl 17 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π = βͺ π½) |
31 | 29, 30 | sseqtrd 3988 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π΄ β βͺ π½) |
32 | | simpr 486 |
. . . . . . . . . 10
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π§ β (π¦ β© ((clsβπ½)βπ΄))) |
33 | 32 | elin2d 4163 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π§ β ((clsβπ½)βπ΄)) |
34 | 5 | adantr 482 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π¦ β π½) |
35 | 32 | elin1d 4162 |
. . . . . . . . 9
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β π§ β π¦) |
36 | 20 | clsndisj 22449 |
. . . . . . . . 9
β’ (((π½ β Top β§ π΄ β βͺ π½
β§ π§ β
((clsβπ½)βπ΄)) β§ (π¦ β π½ β§ π§ β π¦)) β (π¦ β© π΄) β β
) |
37 | 28, 31, 33, 34, 35, 36 | syl32anc 1379 |
. . . . . . . 8
β’
(((((π½ β
(TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β§ π§ β (π¦ β© ((clsβπ½)βπ΄))) β (π¦ β© π΄) β β
) |
38 | 26, 37 | exlimddv 1939 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π¦ β© π΄) β β
) |
39 | | simprl3 1221 |
. . . . . . . . . 10
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) |
40 | 2, 10 | syl 17 |
. . . . . . . . . . . 12
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π½ β Top) |
41 | 2, 13 | syl 17 |
. . . . . . . . . . . . 13
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π = βͺ π½) |
42 | 3, 41 | sseqtrd 3988 |
. . . . . . . . . . . 12
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π΄ β βͺ π½) |
43 | 20 | sscls 22430 |
. . . . . . . . . . . 12
β’ ((π½ β Top β§ π΄ β βͺ π½)
β π΄ β
((clsβπ½)βπ΄)) |
44 | 40, 42, 43 | syl2anc 585 |
. . . . . . . . . . 11
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π΄ β ((clsβπ½)βπ΄)) |
45 | 44 | sscond 4105 |
. . . . . . . . . 10
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π β ((clsβπ½)βπ΄)) β (π β π΄)) |
46 | 39, 45 | sstrd 3958 |
. . . . . . . . 9
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π₯ β© π¦) β (π β π΄)) |
47 | | ssv 3972 |
. . . . . . . . . 10
β’ π β V |
48 | | ssdif 4103 |
. . . . . . . . . 10
β’ (π β V β (π β π΄) β (V β π΄)) |
49 | 47, 48 | ax-mp 5 |
. . . . . . . . 9
β’ (π β π΄) β (V β π΄) |
50 | 46, 49 | sstrdi 3960 |
. . . . . . . 8
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β (π₯ β© π¦) β (V β π΄)) |
51 | | disj2 4421 |
. . . . . . . 8
β’ (((π₯ β© π¦) β© π΄) = β
β (π₯ β© π¦) β (V β π΄)) |
52 | 50, 51 | sylibr 233 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β ((π₯ β© π¦) β© π΄) = β
) |
53 | | simprr 772 |
. . . . . . . 8
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β ((clsβπ½)βπ΄) β (π₯ βͺ π¦)) |
54 | 44, 53 | sstrd 3958 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β π΄ β (π₯ βͺ π¦)) |
55 | 2, 3, 4, 5, 23, 38, 52, 54 | nconnsubb 22797 |
. . . . . 6
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β§ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) β Β¬ (π½ βΎt π΄) β Conn) |
56 | 55 | expr 458 |
. . . . 5
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ ((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄)))) β (((clsβπ½)βπ΄) β (π₯ βͺ π¦) β Β¬ (π½ βΎt π΄) β Conn)) |
57 | 1, 56 | mt2d 136 |
. . . 4
β’ ((((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β§ ((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄)))) β Β¬ ((clsβπ½)βπ΄) β (π₯ βͺ π¦)) |
58 | 57 | ex 414 |
. . 3
β’ (((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β§ (π₯ β π½ β§ π¦ β π½)) β (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β Β¬ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) |
59 | 58 | ralrimivva 3194 |
. 2
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β βπ₯ β π½ βπ¦ β π½ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β Β¬ ((clsβπ½)βπ΄) β (π₯ βͺ π¦))) |
60 | | simp1 1137 |
. . 3
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β π½ β (TopOnβπ)) |
61 | 13 | sseq2d 3980 |
. . . . . . 7
β’ (π½ β (TopOnβπ) β (π΄ β π β π΄ β βͺ π½)) |
62 | 61 | biimpa 478 |
. . . . . 6
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β π΄ β βͺ π½) |
63 | 20 | clsss3 22433 |
. . . . . 6
β’ ((π½ β Top β§ π΄ β βͺ π½)
β ((clsβπ½)βπ΄) β βͺ π½) |
64 | 10, 62, 63 | syl2an2r 684 |
. . . . 5
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β ((clsβπ½)βπ΄) β βͺ π½) |
65 | 13 | adantr 482 |
. . . . 5
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β π = βͺ π½) |
66 | 64, 65 | sseqtrrd 3989 |
. . . 4
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β ((clsβπ½)βπ΄) β π) |
67 | 66 | 3adant3 1133 |
. . 3
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β ((clsβπ½)βπ΄) β π) |
68 | | connsub 22795 |
. . 3
β’ ((π½ β (TopOnβπ) β§ ((clsβπ½)βπ΄) β π) β ((π½ βΎt ((clsβπ½)βπ΄)) β Conn β βπ₯ β π½ βπ¦ β π½ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β Β¬ ((clsβπ½)βπ΄) β (π₯ βͺ π¦)))) |
69 | 60, 67, 68 | syl2anc 585 |
. 2
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β ((π½ βΎt ((clsβπ½)βπ΄)) β Conn β βπ₯ β π½ βπ¦ β π½ (((π₯ β© ((clsβπ½)βπ΄)) β β
β§ (π¦ β© ((clsβπ½)βπ΄)) β β
β§ (π₯ β© π¦) β (π β ((clsβπ½)βπ΄))) β Β¬ ((clsβπ½)βπ΄) β (π₯ βͺ π¦)))) |
70 | 59, 69 | mpbird 257 |
1
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β (π½ βΎt ((clsβπ½)βπ΄)) β Conn) |