Step | Hyp | Ref
| Expression |
1 | | karden.a |
. . . . . . 7
β’ π΄ β V |
2 | | breq1 5109 |
. . . . . . 7
β’ (π€ = π΄ β (π€ β π΄ β π΄ β π΄)) |
3 | 1 | enref 8928 |
. . . . . . 7
β’ π΄ β π΄ |
4 | 1, 2, 3 | ceqsexv2d 3496 |
. . . . . 6
β’
βπ€ π€ β π΄ |
5 | | abn0 4341 |
. . . . . 6
β’ ({π€ β£ π€ β π΄} β β
β βπ€ π€ β π΄) |
6 | 4, 5 | mpbir 230 |
. . . . 5
β’ {π€ β£ π€ β π΄} β β
|
7 | | scott0 9827 |
. . . . . 6
β’ ({π€ β£ π€ β π΄} = β
β {π§ β {π€ β£ π€ β π΄} β£ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)} = β
) |
8 | 7 | necon3bii 2993 |
. . . . 5
β’ ({π€ β£ π€ β π΄} β β
β {π§ β {π€ β£ π€ β π΄} β£ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)} β β
) |
9 | 6, 8 | mpbi 229 |
. . . 4
β’ {π§ β {π€ β£ π€ β π΄} β£ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)} β β
|
10 | | rabn0 4346 |
. . . 4
β’ ({π§ β {π€ β£ π€ β π΄} β£ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)} β β
β βπ§ β {π€ β£ π€ β π΄}βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)) |
11 | 9, 10 | mpbi 229 |
. . 3
β’
βπ§ β
{π€ β£ π€ β π΄}βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦) |
12 | | vex 3448 |
. . . . . . . 8
β’ π§ β V |
13 | | breq1 5109 |
. . . . . . . 8
β’ (π€ = π§ β (π€ β π΄ β π§ β π΄)) |
14 | 12, 13 | elab 3631 |
. . . . . . 7
β’ (π§ β {π€ β£ π€ β π΄} β π§ β π΄) |
15 | | breq1 5109 |
. . . . . . . 8
β’ (π€ = π¦ β (π€ β π΄ β π¦ β π΄)) |
16 | 15 | ralab 3650 |
. . . . . . 7
β’
(βπ¦ β
{π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦) β βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) |
17 | 14, 16 | anbi12i 628 |
. . . . . 6
β’ ((π§ β {π€ β£ π€ β π΄} β§ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)) β (π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦)))) |
18 | | simpl 484 |
. . . . . . . . 9
β’ ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β π§ β π΄) |
19 | 18 | a1i 11 |
. . . . . . . 8
β’ (πΆ = π· β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β π§ β π΄)) |
20 | | karden.c |
. . . . . . . . . . . 12
β’ πΆ = {π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} |
21 | | karden.d |
. . . . . . . . . . . 12
β’ π· = {π₯ β£ (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))} |
22 | 20, 21 | eqeq12i 2751 |
. . . . . . . . . . 11
β’ (πΆ = π· β {π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} = {π₯ β£ (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))}) |
23 | | abbib 2805 |
. . . . . . . . . . 11
β’ ({π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} = {π₯ β£ (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))} β βπ₯((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦))))) |
24 | 22, 23 | bitri 275 |
. . . . . . . . . 10
β’ (πΆ = π· β βπ₯((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦))))) |
25 | | breq1 5109 |
. . . . . . . . . . . . 13
β’ (π₯ = π§ β (π₯ β π΄ β π§ β π΄)) |
26 | | fveq2 6843 |
. . . . . . . . . . . . . . . 16
β’ (π₯ = π§ β (rankβπ₯) = (rankβπ§)) |
27 | 26 | sseq1d 3976 |
. . . . . . . . . . . . . . 15
β’ (π₯ = π§ β ((rankβπ₯) β (rankβπ¦) β (rankβπ§) β (rankβπ¦))) |
28 | 27 | imbi2d 341 |
. . . . . . . . . . . . . 14
β’ (π₯ = π§ β ((π¦ β π΄ β (rankβπ₯) β (rankβπ¦)) β (π¦ β π΄ β (rankβπ§) β (rankβπ¦)))) |
29 | 28 | albidv 1924 |
. . . . . . . . . . . . 13
β’ (π₯ = π§ β (βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)) β βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦)))) |
30 | 25, 29 | anbi12d 632 |
. . . . . . . . . . . 12
β’ (π₯ = π§ β ((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))))) |
31 | | breq1 5109 |
. . . . . . . . . . . . 13
β’ (π₯ = π§ β (π₯ β π΅ β π§ β π΅)) |
32 | 27 | imbi2d 341 |
. . . . . . . . . . . . . 14
β’ (π₯ = π§ β ((π¦ β π΅ β (rankβπ₯) β (rankβπ¦)) β (π¦ β π΅ β (rankβπ§) β (rankβπ¦)))) |
33 | 32 | albidv 1924 |
. . . . . . . . . . . . 13
β’ (π₯ = π§ β (βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)) β βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦)))) |
34 | 31, 33 | anbi12d 632 |
. . . . . . . . . . . 12
β’ (π₯ = π§ β ((π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦))) β (π§ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦))))) |
35 | 30, 34 | bibi12d 346 |
. . . . . . . . . . 11
β’ (π₯ = π§ β (((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))) β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β (π§ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦)))))) |
36 | 35 | spvv 2001 |
. . . . . . . . . 10
β’
(βπ₯((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))) β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β (π§ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦))))) |
37 | 24, 36 | sylbi 216 |
. . . . . . . . 9
β’ (πΆ = π· β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β (π§ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦))))) |
38 | | simpl 484 |
. . . . . . . . 9
β’ ((π§ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ§) β (rankβπ¦))) β π§ β π΅) |
39 | 37, 38 | syl6bi 253 |
. . . . . . . 8
β’ (πΆ = π· β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β π§ β π΅)) |
40 | 19, 39 | jcad 514 |
. . . . . . 7
β’ (πΆ = π· β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β (π§ β π΄ β§ π§ β π΅))) |
41 | | ensym 8946 |
. . . . . . . 8
β’ (π§ β π΄ β π΄ β π§) |
42 | | entr 8949 |
. . . . . . . 8
β’ ((π΄ β π§ β§ π§ β π΅) β π΄ β π΅) |
43 | 41, 42 | sylan 581 |
. . . . . . 7
β’ ((π§ β π΄ β§ π§ β π΅) β π΄ β π΅) |
44 | 40, 43 | syl6 35 |
. . . . . 6
β’ (πΆ = π· β ((π§ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ§) β (rankβπ¦))) β π΄ β π΅)) |
45 | 17, 44 | biimtrid 241 |
. . . . 5
β’ (πΆ = π· β ((π§ β {π€ β£ π€ β π΄} β§ βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦)) β π΄ β π΅)) |
46 | 45 | expd 417 |
. . . 4
β’ (πΆ = π· β (π§ β {π€ β£ π€ β π΄} β (βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦) β π΄ β π΅))) |
47 | 46 | rexlimdv 3147 |
. . 3
β’ (πΆ = π· β (βπ§ β {π€ β£ π€ β π΄}βπ¦ β {π€ β£ π€ β π΄} (rankβπ§) β (rankβπ¦) β π΄ β π΅)) |
48 | 11, 47 | mpi 20 |
. 2
β’ (πΆ = π· β π΄ β π΅) |
49 | | enen2 9065 |
. . . . 5
β’ (π΄ β π΅ β (π₯ β π΄ β π₯ β π΅)) |
50 | | enen2 9065 |
. . . . . . 7
β’ (π΄ β π΅ β (π¦ β π΄ β π¦ β π΅)) |
51 | 50 | imbi1d 342 |
. . . . . 6
β’ (π΄ β π΅ β ((π¦ β π΄ β (rankβπ₯) β (rankβπ¦)) β (π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))) |
52 | 51 | albidv 1924 |
. . . . 5
β’ (π΄ β π΅ β (βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)) β βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))) |
53 | 49, 52 | anbi12d 632 |
. . . 4
β’ (π΄ β π΅ β ((π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦))) β (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦))))) |
54 | 53 | abbidv 2802 |
. . 3
β’ (π΄ β π΅ β {π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} = {π₯ β£ (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))}) |
55 | 54, 20, 21 | 3eqtr4g 2798 |
. 2
β’ (π΄ β π΅ β πΆ = π·) |
56 | 48, 55 | impbii 208 |
1
β’ (πΆ = π· β π΄ β π΅) |