Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( 𝑥 = ∅ → 𝑥 = ∅ ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐴 ↑o 𝑥 ) = ( 𝐴 ↑o ∅ ) ) |
3 |
1 2
|
sseq12d |
⊢ ( 𝑥 = ∅ → ( 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ↔ ∅ ⊆ ( 𝐴 ↑o ∅ ) ) ) |
4 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
5 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 ↑o 𝑥 ) = ( 𝐴 ↑o 𝑦 ) ) |
6 |
4 5
|
sseq12d |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ↔ 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) ) ) |
7 |
|
id |
⊢ ( 𝑥 = suc 𝑦 → 𝑥 = suc 𝑦 ) |
8 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 ↑o 𝑥 ) = ( 𝐴 ↑o suc 𝑦 ) ) |
9 |
7 8
|
sseq12d |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ↔ suc 𝑦 ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) |
10 |
|
id |
⊢ ( 𝑥 = 𝐵 → 𝑥 = 𝐵 ) |
11 |
|
oveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐴 ↑o 𝑥 ) = ( 𝐴 ↑o 𝐵 ) ) |
12 |
10 11
|
sseq12d |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ↔ 𝐵 ⊆ ( 𝐴 ↑o 𝐵 ) ) ) |
13 |
|
0ss |
⊢ ∅ ⊆ ( 𝐴 ↑o ∅ ) |
14 |
13
|
a1i |
⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ∅ ⊆ ( 𝐴 ↑o ∅ ) ) |
15 |
|
eloni |
⊢ ( 𝑦 ∈ On → Ord 𝑦 ) |
16 |
|
eldifi |
⊢ ( 𝐴 ∈ ( On ∖ 2o ) → 𝐴 ∈ On ) |
17 |
|
oecl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ↑o 𝑦 ) ∈ On ) |
18 |
16 17
|
sylan |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( 𝐴 ↑o 𝑦 ) ∈ On ) |
19 |
|
eloni |
⊢ ( ( 𝐴 ↑o 𝑦 ) ∈ On → Ord ( 𝐴 ↑o 𝑦 ) ) |
20 |
18 19
|
syl |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → Ord ( 𝐴 ↑o 𝑦 ) ) |
21 |
|
ordsucsssuc |
⊢ ( ( Ord 𝑦 ∧ Ord ( 𝐴 ↑o 𝑦 ) ) → ( 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) ↔ suc 𝑦 ⊆ suc ( 𝐴 ↑o 𝑦 ) ) ) |
22 |
15 20 21
|
syl2an2 |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) ↔ suc 𝑦 ⊆ suc ( 𝐴 ↑o 𝑦 ) ) ) |
23 |
|
suceloni |
⊢ ( 𝑦 ∈ On → suc 𝑦 ∈ On ) |
24 |
|
oecl |
⊢ ( ( 𝐴 ∈ On ∧ suc 𝑦 ∈ On ) → ( 𝐴 ↑o suc 𝑦 ) ∈ On ) |
25 |
16 23 24
|
syl2an |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( 𝐴 ↑o suc 𝑦 ) ∈ On ) |
26 |
|
eloni |
⊢ ( ( 𝐴 ↑o suc 𝑦 ) ∈ On → Ord ( 𝐴 ↑o suc 𝑦 ) ) |
27 |
25 26
|
syl |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → Ord ( 𝐴 ↑o suc 𝑦 ) ) |
28 |
|
id |
⊢ ( 𝐴 ∈ ( On ∖ 2o ) → 𝐴 ∈ ( On ∖ 2o ) ) |
29 |
|
vex |
⊢ 𝑦 ∈ V |
30 |
29
|
sucid |
⊢ 𝑦 ∈ suc 𝑦 |
31 |
|
oeordi |
⊢ ( ( suc 𝑦 ∈ On ∧ 𝐴 ∈ ( On ∖ 2o ) ) → ( 𝑦 ∈ suc 𝑦 → ( 𝐴 ↑o 𝑦 ) ∈ ( 𝐴 ↑o suc 𝑦 ) ) ) |
32 |
30 31
|
mpi |
⊢ ( ( suc 𝑦 ∈ On ∧ 𝐴 ∈ ( On ∖ 2o ) ) → ( 𝐴 ↑o 𝑦 ) ∈ ( 𝐴 ↑o suc 𝑦 ) ) |
33 |
23 28 32
|
syl2anr |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( 𝐴 ↑o 𝑦 ) ∈ ( 𝐴 ↑o suc 𝑦 ) ) |
34 |
|
ordsucss |
⊢ ( Ord ( 𝐴 ↑o suc 𝑦 ) → ( ( 𝐴 ↑o 𝑦 ) ∈ ( 𝐴 ↑o suc 𝑦 ) → suc ( 𝐴 ↑o 𝑦 ) ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) |
35 |
27 33 34
|
sylc |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → suc ( 𝐴 ↑o 𝑦 ) ⊆ ( 𝐴 ↑o suc 𝑦 ) ) |
36 |
|
sstr2 |
⊢ ( suc 𝑦 ⊆ suc ( 𝐴 ↑o 𝑦 ) → ( suc ( 𝐴 ↑o 𝑦 ) ⊆ ( 𝐴 ↑o suc 𝑦 ) → suc 𝑦 ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) |
37 |
35 36
|
syl5com |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( suc 𝑦 ⊆ suc ( 𝐴 ↑o 𝑦 ) → suc 𝑦 ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) |
38 |
22 37
|
sylbid |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝑦 ∈ On ) → ( 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → suc 𝑦 ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) |
39 |
38
|
expcom |
⊢ ( 𝑦 ∈ On → ( 𝐴 ∈ ( On ∖ 2o ) → ( 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → suc 𝑦 ⊆ ( 𝐴 ↑o suc 𝑦 ) ) ) ) |
40 |
|
dif20el |
⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ∅ ∈ 𝐴 ) |
41 |
16 40
|
jca |
⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) |
42 |
|
ss2iun |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) |
43 |
|
limuni |
⊢ ( Lim 𝑥 → 𝑥 = ∪ 𝑥 ) |
44 |
|
uniiun |
⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 |
45 |
43 44
|
eqtrdi |
⊢ ( Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 ) |
46 |
45
|
adantr |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 ) |
47 |
|
vex |
⊢ 𝑥 ∈ V |
48 |
|
oelim |
⊢ ( ( ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝐴 ↑o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) |
49 |
47 48
|
mpanlr1 |
⊢ ( ( ( 𝐴 ∈ On ∧ Lim 𝑥 ) ∧ ∅ ∈ 𝐴 ) → ( 𝐴 ↑o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) |
50 |
49
|
anasss |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ ∅ ∈ 𝐴 ) ) → ( 𝐴 ↑o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) |
51 |
50
|
an12s |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ( 𝐴 ↑o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) |
52 |
46 51
|
sseq12d |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ( 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 ( 𝐴 ↑o 𝑦 ) ) ) |
53 |
42 52
|
syl5ibr |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ) ) |
54 |
53
|
ex |
⊢ ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ ∅ ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ) ) ) |
55 |
41 54
|
syl5 |
⊢ ( Lim 𝑥 → ( 𝐴 ∈ ( On ∖ 2o ) → ( ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ ( 𝐴 ↑o 𝑦 ) → 𝑥 ⊆ ( 𝐴 ↑o 𝑥 ) ) ) ) |
56 |
3 6 9 12 14 39 55
|
tfinds3 |
⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ ( On ∖ 2o ) → 𝐵 ⊆ ( 𝐴 ↑o 𝐵 ) ) ) |
57 |
56
|
impcom |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝐵 ∈ On ) → 𝐵 ⊆ ( 𝐴 ↑o 𝐵 ) ) |