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