Step |
Hyp |
Ref |
Expression |
1 |
|
tfindsg.1 |
|- ( x = B -> ( ph <-> ps ) ) |
2 |
|
tfindsg.2 |
|- ( x = y -> ( ph <-> ch ) ) |
3 |
|
tfindsg.3 |
|- ( x = suc y -> ( ph <-> th ) ) |
4 |
|
tfindsg.4 |
|- ( x = A -> ( ph <-> ta ) ) |
5 |
|
tfindsg.5 |
|- ( B e. On -> ps ) |
6 |
|
tfindsg.6 |
|- ( ( ( y e. On /\ B e. On ) /\ B C_ y ) -> ( ch -> th ) ) |
7 |
|
tfindsg.7 |
|- ( ( ( Lim x /\ B e. On ) /\ B C_ x ) -> ( A. y e. x ( B C_ y -> ch ) -> ph ) ) |
8 |
|
sseq2 |
|- ( x = (/) -> ( B C_ x <-> B C_ (/) ) ) |
9 |
8
|
adantl |
|- ( ( B = (/) /\ x = (/) ) -> ( B C_ x <-> B C_ (/) ) ) |
10 |
|
eqeq2 |
|- ( B = (/) -> ( x = B <-> x = (/) ) ) |
11 |
10 1
|
syl6bir |
|- ( B = (/) -> ( x = (/) -> ( ph <-> ps ) ) ) |
12 |
11
|
imp |
|- ( ( B = (/) /\ x = (/) ) -> ( ph <-> ps ) ) |
13 |
9 12
|
imbi12d |
|- ( ( B = (/) /\ x = (/) ) -> ( ( B C_ x -> ph ) <-> ( B C_ (/) -> ps ) ) ) |
14 |
8
|
imbi1d |
|- ( x = (/) -> ( ( B C_ x -> ph ) <-> ( B C_ (/) -> ph ) ) ) |
15 |
|
ss0 |
|- ( B C_ (/) -> B = (/) ) |
16 |
15
|
con3i |
|- ( -. B = (/) -> -. B C_ (/) ) |
17 |
16
|
pm2.21d |
|- ( -. B = (/) -> ( B C_ (/) -> ( ph <-> ps ) ) ) |
18 |
17
|
pm5.74d |
|- ( -. B = (/) -> ( ( B C_ (/) -> ph ) <-> ( B C_ (/) -> ps ) ) ) |
19 |
14 18
|
sylan9bbr |
|- ( ( -. B = (/) /\ x = (/) ) -> ( ( B C_ x -> ph ) <-> ( B C_ (/) -> ps ) ) ) |
20 |
13 19
|
pm2.61ian |
|- ( x = (/) -> ( ( B C_ x -> ph ) <-> ( B C_ (/) -> ps ) ) ) |
21 |
20
|
imbi2d |
|- ( x = (/) -> ( ( B e. On -> ( B C_ x -> ph ) ) <-> ( B e. On -> ( B C_ (/) -> ps ) ) ) ) |
22 |
|
sseq2 |
|- ( x = y -> ( B C_ x <-> B C_ y ) ) |
23 |
22 2
|
imbi12d |
|- ( x = y -> ( ( B C_ x -> ph ) <-> ( B C_ y -> ch ) ) ) |
24 |
23
|
imbi2d |
|- ( x = y -> ( ( B e. On -> ( B C_ x -> ph ) ) <-> ( B e. On -> ( B C_ y -> ch ) ) ) ) |
25 |
|
sseq2 |
|- ( x = suc y -> ( B C_ x <-> B C_ suc y ) ) |
26 |
25 3
|
imbi12d |
|- ( x = suc y -> ( ( B C_ x -> ph ) <-> ( B C_ suc y -> th ) ) ) |
27 |
26
|
imbi2d |
|- ( x = suc y -> ( ( B e. On -> ( B C_ x -> ph ) ) <-> ( B e. On -> ( B C_ suc y -> th ) ) ) ) |
28 |
|
sseq2 |
|- ( x = A -> ( B C_ x <-> B C_ A ) ) |
29 |
28 4
|
imbi12d |
|- ( x = A -> ( ( B C_ x -> ph ) <-> ( B C_ A -> ta ) ) ) |
30 |
29
|
imbi2d |
|- ( x = A -> ( ( B e. On -> ( B C_ x -> ph ) ) <-> ( B e. On -> ( B C_ A -> ta ) ) ) ) |
31 |
5
|
a1d |
|- ( B e. On -> ( B C_ (/) -> ps ) ) |
32 |
|
vex |
|- y e. _V |
33 |
32
|
sucex |
|- suc y e. _V |
34 |
33
|
eqvinc |
|- ( suc y = B <-> E. x ( x = suc y /\ x = B ) ) |
35 |
5 1
|
syl5ibr |
|- ( x = B -> ( B e. On -> ph ) ) |
36 |
3
|
biimpd |
|- ( x = suc y -> ( ph -> th ) ) |
37 |
35 36
|
sylan9r |
|- ( ( x = suc y /\ x = B ) -> ( B e. On -> th ) ) |
38 |
37
|
exlimiv |
|- ( E. x ( x = suc y /\ x = B ) -> ( B e. On -> th ) ) |
39 |
34 38
|
sylbi |
|- ( suc y = B -> ( B e. On -> th ) ) |
40 |
39
|
eqcoms |
|- ( B = suc y -> ( B e. On -> th ) ) |
41 |
40
|
imim2i |
|- ( ( B C_ suc y -> B = suc y ) -> ( B C_ suc y -> ( B e. On -> th ) ) ) |
42 |
41
|
a1d |
|- ( ( B C_ suc y -> B = suc y ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> ( B e. On -> th ) ) ) ) |
43 |
42
|
com4r |
|- ( B e. On -> ( ( B C_ suc y -> B = suc y ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
44 |
43
|
adantl |
|- ( ( y e. On /\ B e. On ) -> ( ( B C_ suc y -> B = suc y ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
45 |
|
df-ne |
|- ( B =/= suc y <-> -. B = suc y ) |
46 |
45
|
anbi2i |
|- ( ( B C_ suc y /\ B =/= suc y ) <-> ( B C_ suc y /\ -. B = suc y ) ) |
47 |
|
annim |
|- ( ( B C_ suc y /\ -. B = suc y ) <-> -. ( B C_ suc y -> B = suc y ) ) |
48 |
46 47
|
bitri |
|- ( ( B C_ suc y /\ B =/= suc y ) <-> -. ( B C_ suc y -> B = suc y ) ) |
49 |
|
onsssuc |
|- ( ( B e. On /\ y e. On ) -> ( B C_ y <-> B e. suc y ) ) |
50 |
|
suceloni |
|- ( y e. On -> suc y e. On ) |
51 |
|
onelpss |
|- ( ( B e. On /\ suc y e. On ) -> ( B e. suc y <-> ( B C_ suc y /\ B =/= suc y ) ) ) |
52 |
50 51
|
sylan2 |
|- ( ( B e. On /\ y e. On ) -> ( B e. suc y <-> ( B C_ suc y /\ B =/= suc y ) ) ) |
53 |
49 52
|
bitrd |
|- ( ( B e. On /\ y e. On ) -> ( B C_ y <-> ( B C_ suc y /\ B =/= suc y ) ) ) |
54 |
53
|
ancoms |
|- ( ( y e. On /\ B e. On ) -> ( B C_ y <-> ( B C_ suc y /\ B =/= suc y ) ) ) |
55 |
6
|
ex |
|- ( ( y e. On /\ B e. On ) -> ( B C_ y -> ( ch -> th ) ) ) |
56 |
55
|
a1ddd |
|- ( ( y e. On /\ B e. On ) -> ( B C_ y -> ( ch -> ( B C_ suc y -> th ) ) ) ) |
57 |
56
|
a2d |
|- ( ( y e. On /\ B e. On ) -> ( ( B C_ y -> ch ) -> ( B C_ y -> ( B C_ suc y -> th ) ) ) ) |
58 |
57
|
com23 |
|- ( ( y e. On /\ B e. On ) -> ( B C_ y -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
59 |
54 58
|
sylbird |
|- ( ( y e. On /\ B e. On ) -> ( ( B C_ suc y /\ B =/= suc y ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
60 |
48 59
|
syl5bir |
|- ( ( y e. On /\ B e. On ) -> ( -. ( B C_ suc y -> B = suc y ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
61 |
44 60
|
pm2.61d |
|- ( ( y e. On /\ B e. On ) -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) |
62 |
61
|
ex |
|- ( y e. On -> ( B e. On -> ( ( B C_ y -> ch ) -> ( B C_ suc y -> th ) ) ) ) |
63 |
62
|
a2d |
|- ( y e. On -> ( ( B e. On -> ( B C_ y -> ch ) ) -> ( B e. On -> ( B C_ suc y -> th ) ) ) ) |
64 |
|
pm2.27 |
|- ( B e. On -> ( ( B e. On -> ( B C_ y -> ch ) ) -> ( B C_ y -> ch ) ) ) |
65 |
64
|
ralimdv |
|- ( B e. On -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> A. y e. x ( B C_ y -> ch ) ) ) |
66 |
65
|
ad2antlr |
|- ( ( ( Lim x /\ B e. On ) /\ B C_ x ) -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> A. y e. x ( B C_ y -> ch ) ) ) |
67 |
66 7
|
syld |
|- ( ( ( Lim x /\ B e. On ) /\ B C_ x ) -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> ph ) ) |
68 |
67
|
exp31 |
|- ( Lim x -> ( B e. On -> ( B C_ x -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> ph ) ) ) ) |
69 |
68
|
com3l |
|- ( B e. On -> ( B C_ x -> ( Lim x -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> ph ) ) ) ) |
70 |
69
|
com4t |
|- ( Lim x -> ( A. y e. x ( B e. On -> ( B C_ y -> ch ) ) -> ( B e. On -> ( B C_ x -> ph ) ) ) ) |
71 |
21 24 27 30 31 63 70
|
tfinds |
|- ( A e. On -> ( B e. On -> ( B C_ A -> ta ) ) ) |
72 |
71
|
imp31 |
|- ( ( ( A e. On /\ B e. On ) /\ B C_ A ) -> ta ) |