Step |
Hyp |
Ref |
Expression |
1 |
|
oawordeulem.1 |
|- A e. On |
2 |
|
oawordeulem.2 |
|- B e. On |
3 |
|
oawordeulem.3 |
|- S = { y e. On | B C_ ( A +o y ) } |
4 |
3
|
ssrab3 |
|- S C_ On |
5 |
|
oaword2 |
|- ( ( B e. On /\ A e. On ) -> B C_ ( A +o B ) ) |
6 |
2 1 5
|
mp2an |
|- B C_ ( A +o B ) |
7 |
|
oveq2 |
|- ( y = B -> ( A +o y ) = ( A +o B ) ) |
8 |
7
|
sseq2d |
|- ( y = B -> ( B C_ ( A +o y ) <-> B C_ ( A +o B ) ) ) |
9 |
8 3
|
elrab2 |
|- ( B e. S <-> ( B e. On /\ B C_ ( A +o B ) ) ) |
10 |
2 6 9
|
mpbir2an |
|- B e. S |
11 |
10
|
ne0ii |
|- S =/= (/) |
12 |
|
oninton |
|- ( ( S C_ On /\ S =/= (/) ) -> |^| S e. On ) |
13 |
4 11 12
|
mp2an |
|- |^| S e. On |
14 |
|
onzsl |
|- ( |^| S e. On <-> ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) ) |
15 |
13 14
|
mpbi |
|- ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) |
16 |
|
oveq2 |
|- ( |^| S = (/) -> ( A +o |^| S ) = ( A +o (/) ) ) |
17 |
|
oa0 |
|- ( A e. On -> ( A +o (/) ) = A ) |
18 |
1 17
|
ax-mp |
|- ( A +o (/) ) = A |
19 |
16 18
|
eqtrdi |
|- ( |^| S = (/) -> ( A +o |^| S ) = A ) |
20 |
19
|
sseq1d |
|- ( |^| S = (/) -> ( ( A +o |^| S ) C_ B <-> A C_ B ) ) |
21 |
20
|
biimprd |
|- ( |^| S = (/) -> ( A C_ B -> ( A +o |^| S ) C_ B ) ) |
22 |
|
oveq2 |
|- ( |^| S = suc z -> ( A +o |^| S ) = ( A +o suc z ) ) |
23 |
|
oasuc |
|- ( ( A e. On /\ z e. On ) -> ( A +o suc z ) = suc ( A +o z ) ) |
24 |
1 23
|
mpan |
|- ( z e. On -> ( A +o suc z ) = suc ( A +o z ) ) |
25 |
22 24
|
sylan9eqr |
|- ( ( z e. On /\ |^| S = suc z ) -> ( A +o |^| S ) = suc ( A +o z ) ) |
26 |
|
vex |
|- z e. _V |
27 |
26
|
sucid |
|- z e. suc z |
28 |
|
eleq2 |
|- ( |^| S = suc z -> ( z e. |^| S <-> z e. suc z ) ) |
29 |
27 28
|
mpbiri |
|- ( |^| S = suc z -> z e. |^| S ) |
30 |
13
|
oneli |
|- ( z e. |^| S -> z e. On ) |
31 |
3
|
inteqi |
|- |^| S = |^| { y e. On | B C_ ( A +o y ) } |
32 |
31
|
eleq2i |
|- ( z e. |^| S <-> z e. |^| { y e. On | B C_ ( A +o y ) } ) |
33 |
|
oveq2 |
|- ( y = z -> ( A +o y ) = ( A +o z ) ) |
34 |
33
|
sseq2d |
|- ( y = z -> ( B C_ ( A +o y ) <-> B C_ ( A +o z ) ) ) |
35 |
34
|
onnminsb |
|- ( z e. On -> ( z e. |^| { y e. On | B C_ ( A +o y ) } -> -. B C_ ( A +o z ) ) ) |
36 |
32 35
|
syl5bi |
|- ( z e. On -> ( z e. |^| S -> -. B C_ ( A +o z ) ) ) |
37 |
|
oacl |
|- ( ( A e. On /\ z e. On ) -> ( A +o z ) e. On ) |
38 |
1 37
|
mpan |
|- ( z e. On -> ( A +o z ) e. On ) |
39 |
|
ontri1 |
|- ( ( B e. On /\ ( A +o z ) e. On ) -> ( B C_ ( A +o z ) <-> -. ( A +o z ) e. B ) ) |
40 |
2 38 39
|
sylancr |
|- ( z e. On -> ( B C_ ( A +o z ) <-> -. ( A +o z ) e. B ) ) |
41 |
40
|
con2bid |
|- ( z e. On -> ( ( A +o z ) e. B <-> -. B C_ ( A +o z ) ) ) |
42 |
36 41
|
sylibrd |
|- ( z e. On -> ( z e. |^| S -> ( A +o z ) e. B ) ) |
43 |
30 42
|
mpcom |
|- ( z e. |^| S -> ( A +o z ) e. B ) |
44 |
2
|
onordi |
|- Ord B |
45 |
|
ordsucss |
|- ( Ord B -> ( ( A +o z ) e. B -> suc ( A +o z ) C_ B ) ) |
46 |
44 45
|
ax-mp |
|- ( ( A +o z ) e. B -> suc ( A +o z ) C_ B ) |
47 |
29 43 46
|
3syl |
|- ( |^| S = suc z -> suc ( A +o z ) C_ B ) |
48 |
47
|
adantl |
|- ( ( z e. On /\ |^| S = suc z ) -> suc ( A +o z ) C_ B ) |
49 |
25 48
|
eqsstrd |
|- ( ( z e. On /\ |^| S = suc z ) -> ( A +o |^| S ) C_ B ) |
50 |
49
|
rexlimiva |
|- ( E. z e. On |^| S = suc z -> ( A +o |^| S ) C_ B ) |
51 |
50
|
a1d |
|- ( E. z e. On |^| S = suc z -> ( A C_ B -> ( A +o |^| S ) C_ B ) ) |
52 |
|
oalim |
|- ( ( A e. On /\ ( |^| S e. _V /\ Lim |^| S ) ) -> ( A +o |^| S ) = U_ z e. |^| S ( A +o z ) ) |
53 |
1 52
|
mpan |
|- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A +o |^| S ) = U_ z e. |^| S ( A +o z ) ) |
54 |
|
iunss |
|- ( U_ z e. |^| S ( A +o z ) C_ B <-> A. z e. |^| S ( A +o z ) C_ B ) |
55 |
2
|
onelssi |
|- ( ( A +o z ) e. B -> ( A +o z ) C_ B ) |
56 |
43 55
|
syl |
|- ( z e. |^| S -> ( A +o z ) C_ B ) |
57 |
54 56
|
mprgbir |
|- U_ z e. |^| S ( A +o z ) C_ B |
58 |
53 57
|
eqsstrdi |
|- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A +o |^| S ) C_ B ) |
59 |
58
|
a1d |
|- ( ( |^| S e. _V /\ Lim |^| S ) -> ( A C_ B -> ( A +o |^| S ) C_ B ) ) |
60 |
21 51 59
|
3jaoi |
|- ( ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) -> ( A C_ B -> ( A +o |^| S ) C_ B ) ) |
61 |
15 60
|
ax-mp |
|- ( A C_ B -> ( A +o |^| S ) C_ B ) |
62 |
8
|
rspcev |
|- ( ( B e. On /\ B C_ ( A +o B ) ) -> E. y e. On B C_ ( A +o y ) ) |
63 |
2 6 62
|
mp2an |
|- E. y e. On B C_ ( A +o y ) |
64 |
|
nfcv |
|- F/_ y B |
65 |
|
nfcv |
|- F/_ y A |
66 |
|
nfcv |
|- F/_ y +o |
67 |
|
nfrab1 |
|- F/_ y { y e. On | B C_ ( A +o y ) } |
68 |
67
|
nfint |
|- F/_ y |^| { y e. On | B C_ ( A +o y ) } |
69 |
65 66 68
|
nfov |
|- F/_ y ( A +o |^| { y e. On | B C_ ( A +o y ) } ) |
70 |
64 69
|
nfss |
|- F/ y B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) |
71 |
|
oveq2 |
|- ( y = |^| { y e. On | B C_ ( A +o y ) } -> ( A +o y ) = ( A +o |^| { y e. On | B C_ ( A +o y ) } ) ) |
72 |
71
|
sseq2d |
|- ( y = |^| { y e. On | B C_ ( A +o y ) } -> ( B C_ ( A +o y ) <-> B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) ) ) |
73 |
70 72
|
onminsb |
|- ( E. y e. On B C_ ( A +o y ) -> B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) ) |
74 |
63 73
|
ax-mp |
|- B C_ ( A +o |^| { y e. On | B C_ ( A +o y ) } ) |
75 |
31
|
oveq2i |
|- ( A +o |^| S ) = ( A +o |^| { y e. On | B C_ ( A +o y ) } ) |
76 |
74 75
|
sseqtrri |
|- B C_ ( A +o |^| S ) |
77 |
|
eqss |
|- ( ( A +o |^| S ) = B <-> ( ( A +o |^| S ) C_ B /\ B C_ ( A +o |^| S ) ) ) |
78 |
61 76 77
|
sylanblrc |
|- ( A C_ B -> ( A +o |^| S ) = B ) |
79 |
|
oveq2 |
|- ( x = |^| S -> ( A +o x ) = ( A +o |^| S ) ) |
80 |
79
|
eqeq1d |
|- ( x = |^| S -> ( ( A +o x ) = B <-> ( A +o |^| S ) = B ) ) |
81 |
80
|
rspcev |
|- ( ( |^| S e. On /\ ( A +o |^| S ) = B ) -> E. x e. On ( A +o x ) = B ) |
82 |
13 78 81
|
sylancr |
|- ( A C_ B -> E. x e. On ( A +o x ) = B ) |
83 |
|
eqtr3 |
|- ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> ( A +o x ) = ( A +o y ) ) |
84 |
|
oacan |
|- ( ( A e. On /\ x e. On /\ y e. On ) -> ( ( A +o x ) = ( A +o y ) <-> x = y ) ) |
85 |
1 84
|
mp3an1 |
|- ( ( x e. On /\ y e. On ) -> ( ( A +o x ) = ( A +o y ) <-> x = y ) ) |
86 |
83 85
|
syl5ib |
|- ( ( x e. On /\ y e. On ) -> ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) ) |
87 |
86
|
rgen2 |
|- A. x e. On A. y e. On ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) |
88 |
|
oveq2 |
|- ( x = y -> ( A +o x ) = ( A +o y ) ) |
89 |
88
|
eqeq1d |
|- ( x = y -> ( ( A +o x ) = B <-> ( A +o y ) = B ) ) |
90 |
89
|
reu4 |
|- ( E! x e. On ( A +o x ) = B <-> ( E. x e. On ( A +o x ) = B /\ A. x e. On A. y e. On ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) ) ) |
91 |
82 87 90
|
sylanblrc |
|- ( A C_ B -> E! x e. On ( A +o x ) = B ) |