Step | Hyp | Ref
| Expression |
1 | | simp1 1137 |
. . . . . 6
β’ ((π½ β Top β§ π΄ β π β§ π β π΄) β π½ β Top) |
2 | | simp2 1138 |
. . . . . 6
β’ ((π½ β Top β§ π΄ β π β§ π β π΄) β π΄ β π) |
3 | | ssel2 3940 |
. . . . . . 7
β’ ((π΄ β π β§ π β π΄) β π β π) |
4 | 3 | 3adant1 1131 |
. . . . . 6
β’ ((π½ β Top β§ π΄ β π β§ π β π΄) β π β π) |
5 | 1, 2, 4 | 3jca 1129 |
. . . . 5
β’ ((π½ β Top β§ π΄ β π β§ π β π΄) β (π½ β Top β§ π΄ β π β§ π β π)) |
6 | | noel 4291 |
. . . . . . . . 9
β’ Β¬
π β
β
|
7 | | eleq2 2823 |
. . . . . . . . 9
β’
(((limPtβπ½)βπ΄) = β
β (π β ((limPtβπ½)βπ΄) β π β β
)) |
8 | 6, 7 | mtbiri 327 |
. . . . . . . 8
β’
(((limPtβπ½)βπ΄) = β
β Β¬ π β ((limPtβπ½)βπ΄)) |
9 | 8 | adantl 483 |
. . . . . . 7
β’ (((π½ β Top β§ π΄ β π β§ π β π) β§ ((limPtβπ½)βπ΄) = β
) β Β¬ π β ((limPtβπ½)βπ΄)) |
10 | | nlpineqsn.x |
. . . . . . . . 9
β’ π = βͺ
π½ |
11 | 10 | islp3 22513 |
. . . . . . . 8
β’ ((π½ β Top β§ π΄ β π β§ π β π) β (π β ((limPtβπ½)βπ΄) β βπ β π½ (π β π β (π β© (π΄ β {π})) β β
))) |
12 | 11 | adantr 482 |
. . . . . . 7
β’ (((π½ β Top β§ π΄ β π β§ π β π) β§ ((limPtβπ½)βπ΄) = β
) β (π β ((limPtβπ½)βπ΄) β βπ β π½ (π β π β (π β© (π΄ β {π})) β β
))) |
13 | 9, 12 | mtbid 324 |
. . . . . 6
β’ (((π½ β Top β§ π΄ β π β§ π β π) β§ ((limPtβπ½)βπ΄) = β
) β Β¬ βπ β π½ (π β π β (π β© (π΄ β {π})) β β
)) |
14 | | nne 2944 |
. . . . . . . . . 10
β’ (Β¬
(π β© (π΄ β {π})) β β
β (π β© (π΄ β {π})) = β
) |
15 | 14 | anbi2i 624 |
. . . . . . . . 9
β’ ((π β π β§ Β¬ (π β© (π΄ β {π})) β β
) β (π β π β§ (π β© (π΄ β {π})) = β
)) |
16 | | annim 405 |
. . . . . . . . 9
β’ ((π β π β§ Β¬ (π β© (π΄ β {π})) β β
) β Β¬ (π β π β (π β© (π΄ β {π})) β β
)) |
17 | 15, 16 | bitr3i 277 |
. . . . . . . 8
β’ ((π β π β§ (π β© (π΄ β {π})) = β
) β Β¬ (π β π β (π β© (π΄ β {π})) β β
)) |
18 | 17 | rexbii 3094 |
. . . . . . 7
β’
(βπ β
π½ (π β π β§ (π β© (π΄ β {π})) = β
) β βπ β π½ Β¬ (π β π β (π β© (π΄ β {π})) β β
)) |
19 | | rexnal 3100 |
. . . . . . 7
β’
(βπ β
π½ Β¬ (π β π β (π β© (π΄ β {π})) β β
) β Β¬ βπ β π½ (π β π β (π β© (π΄ β {π})) β β
)) |
20 | 18, 19 | bitri 275 |
. . . . . 6
β’
(βπ β
π½ (π β π β§ (π β© (π΄ β {π})) = β
) β Β¬ βπ β π½ (π β π β (π β© (π΄ β {π})) β β
)) |
21 | 13, 20 | sylibr 233 |
. . . . 5
β’ (((π½ β Top β§ π΄ β π β§ π β π) β§ ((limPtβπ½)βπ΄) = β
) β βπ β π½ (π β π β§ (π β© (π΄ β {π})) = β
)) |
22 | 5, 21 | sylan 581 |
. . . 4
β’ (((π½ β Top β§ π΄ β π β§ π β π΄) β§ ((limPtβπ½)βπ΄) = β
) β βπ β π½ (π β π β§ (π β© (π΄ β {π})) = β
)) |
23 | | indif2 4231 |
. . . . . . . . . . . 12
β’ (π β© (π΄ β {π})) = ((π β© π΄) β {π}) |
24 | 23 | eqeq1i 2738 |
. . . . . . . . . . 11
β’ ((π β© (π΄ β {π})) = β
β ((π β© π΄) β {π}) = β
) |
25 | | ssdif0 4324 |
. . . . . . . . . . 11
β’ ((π β© π΄) β {π} β ((π β© π΄) β {π}) = β
) |
26 | 24, 25 | bitr4i 278 |
. . . . . . . . . 10
β’ ((π β© (π΄ β {π})) = β
β (π β© π΄) β {π}) |
27 | | elin 3927 |
. . . . . . . . . . 11
β’ (π β (π β© π΄) β (π β π β§ π β π΄)) |
28 | | sssn 4787 |
. . . . . . . . . . . 12
β’ ((π β© π΄) β {π} β ((π β© π΄) = β
β¨ (π β© π΄) = {π})) |
29 | | n0i 4294 |
. . . . . . . . . . . . 13
β’ (π β (π β© π΄) β Β¬ (π β© π΄) = β
) |
30 | | biorf 936 |
. . . . . . . . . . . . 13
β’ (Β¬
(π β© π΄) = β
β ((π β© π΄) = {π} β ((π β© π΄) = β
β¨ (π β© π΄) = {π}))) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . 12
β’ (π β (π β© π΄) β ((π β© π΄) = {π} β ((π β© π΄) = β
β¨ (π β© π΄) = {π}))) |
32 | 28, 31 | bitr4id 290 |
. . . . . . . . . . 11
β’ (π β (π β© π΄) β ((π β© π΄) β {π} β (π β© π΄) = {π})) |
33 | 27, 32 | sylbir 234 |
. . . . . . . . . 10
β’ ((π β π β§ π β π΄) β ((π β© π΄) β {π} β (π β© π΄) = {π})) |
34 | 26, 33 | bitrid 283 |
. . . . . . . . 9
β’ ((π β π β§ π β π΄) β ((π β© (π΄ β {π})) = β
β (π β© π΄) = {π})) |
35 | 34 | ancoms 460 |
. . . . . . . 8
β’ ((π β π΄ β§ π β π) β ((π β© (π΄ β {π})) = β
β (π β© π΄) = {π})) |
36 | 35 | pm5.32da 580 |
. . . . . . 7
β’ (π β π΄ β ((π β π β§ (π β© (π΄ β {π})) = β
) β (π β π β§ (π β© π΄) = {π}))) |
37 | 36 | rexbidv 3172 |
. . . . . 6
β’ (π β π΄ β (βπ β π½ (π β π β§ (π β© (π΄ β {π})) = β
) β βπ β π½ (π β π β§ (π β© π΄) = {π}))) |
38 | 37 | 3ad2ant3 1136 |
. . . . 5
β’ ((π½ β Top β§ π΄ β π β§ π β π΄) β (βπ β π½ (π β π β§ (π β© (π΄ β {π})) = β
) β βπ β π½ (π β π β§ (π β© π΄) = {π}))) |
39 | 38 | adantr 482 |
. . . 4
β’ (((π½ β Top β§ π΄ β π β§ π β π΄) β§ ((limPtβπ½)βπ΄) = β
) β (βπ β π½ (π β π β§ (π β© (π΄ β {π})) = β
) β βπ β π½ (π β π β§ (π β© π΄) = {π}))) |
40 | 22, 39 | mpbid 231 |
. . 3
β’ (((π½ β Top β§ π΄ β π β§ π β π΄) β§ ((limPtβπ½)βπ΄) = β
) β βπ β π½ (π β π β§ (π β© π΄) = {π})) |
41 | 40 | 3an1rs 1360 |
. 2
β’ (((π½ β Top β§ π΄ β π β§ ((limPtβπ½)βπ΄) = β
) β§ π β π΄) β βπ β π½ (π β π β§ (π β© π΄) = {π})) |
42 | 41 | ralrimiva 3140 |
1
β’ ((π½ β Top β§ π΄ β π β§ ((limPtβπ½)βπ΄) = β
) β βπ β π΄ βπ β π½ (π β π β§ (π β© π΄) = {π})) |