Step |
Hyp |
Ref |
Expression |
1 |
|
orduni |
⊢ ( Ord 𝐴 → Ord ∪ 𝐴 ) |
2 |
|
unizlim |
⊢ ( Ord ∪ 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 ↔ ( ∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴 ) ) ) |
3 |
|
uni0b |
⊢ ( ∪ 𝐴 = ∅ ↔ 𝐴 ⊆ { ∅ } ) |
4 |
3
|
orbi1i |
⊢ ( ( ∪ 𝐴 = ∅ ∨ Lim ∪ 𝐴 ) ↔ ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) ) |
5 |
2 4
|
bitrdi |
⊢ ( Ord ∪ 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 ↔ ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) ) ) |
6 |
5
|
biimpd |
⊢ ( Ord ∪ 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 → ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) ) ) |
7 |
1 6
|
syl |
⊢ ( Ord 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 → ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) ) ) |
8 |
|
sssn |
⊢ ( 𝐴 ⊆ { ∅ } ↔ ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) ) |
9 |
|
0ntop |
⊢ ¬ ∅ ∈ Top |
10 |
|
cmptop |
⊢ ( ∅ ∈ Comp → ∅ ∈ Top ) |
11 |
9 10
|
mto |
⊢ ¬ ∅ ∈ Comp |
12 |
|
eleq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∈ Comp ↔ ∅ ∈ Comp ) ) |
13 |
11 12
|
mtbiri |
⊢ ( 𝐴 = ∅ → ¬ 𝐴 ∈ Comp ) |
14 |
13
|
pm2.21d |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) |
15 |
|
id |
⊢ ( 𝐴 = { ∅ } → 𝐴 = { ∅ } ) |
16 |
|
df1o2 |
⊢ 1o = { ∅ } |
17 |
15 16
|
eqtr4di |
⊢ ( 𝐴 = { ∅ } → 𝐴 = 1o ) |
18 |
17
|
a1d |
⊢ ( 𝐴 = { ∅ } → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) |
19 |
14 18
|
jaoi |
⊢ ( ( 𝐴 = ∅ ∨ 𝐴 = { ∅ } ) → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) |
20 |
8 19
|
sylbi |
⊢ ( 𝐴 ⊆ { ∅ } → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) |
21 |
20
|
a1i |
⊢ ( Ord 𝐴 → ( 𝐴 ⊆ { ∅ } → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) ) |
22 |
|
ordtop |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ Top ↔ 𝐴 ≠ ∪ 𝐴 ) ) |
23 |
22
|
biimpd |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ Top → 𝐴 ≠ ∪ 𝐴 ) ) |
24 |
23
|
necon2bd |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Top ) ) |
25 |
|
cmptop |
⊢ ( 𝐴 ∈ Comp → 𝐴 ∈ Top ) |
26 |
25
|
con3i |
⊢ ( ¬ 𝐴 ∈ Top → ¬ 𝐴 ∈ Comp ) |
27 |
24 26
|
syl6 |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 → ¬ 𝐴 ∈ Comp ) ) |
28 |
27
|
a1dd |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 → ( Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp ) ) ) |
29 |
|
limsucncmp |
⊢ ( Lim ∪ 𝐴 → ¬ suc ∪ 𝐴 ∈ Comp ) |
30 |
|
eleq1 |
⊢ ( 𝐴 = suc ∪ 𝐴 → ( 𝐴 ∈ Comp ↔ suc ∪ 𝐴 ∈ Comp ) ) |
31 |
30
|
notbid |
⊢ ( 𝐴 = suc ∪ 𝐴 → ( ¬ 𝐴 ∈ Comp ↔ ¬ suc ∪ 𝐴 ∈ Comp ) ) |
32 |
29 31
|
syl5ibr |
⊢ ( 𝐴 = suc ∪ 𝐴 → ( Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp ) ) |
33 |
32
|
a1i |
⊢ ( Ord 𝐴 → ( 𝐴 = suc ∪ 𝐴 → ( Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp ) ) ) |
34 |
|
orduniorsuc |
⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) ) |
35 |
28 33 34
|
mpjaod |
⊢ ( Ord 𝐴 → ( Lim ∪ 𝐴 → ¬ 𝐴 ∈ Comp ) ) |
36 |
|
pm2.21 |
⊢ ( ¬ 𝐴 ∈ Comp → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) |
37 |
35 36
|
syl6 |
⊢ ( Ord 𝐴 → ( Lim ∪ 𝐴 → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) ) |
38 |
21 37
|
jaod |
⊢ ( Ord 𝐴 → ( ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) → ( 𝐴 ∈ Comp → 𝐴 = 1o ) ) ) |
39 |
38
|
com23 |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ Comp → ( ( 𝐴 ⊆ { ∅ } ∨ Lim ∪ 𝐴 ) → 𝐴 = 1o ) ) ) |
40 |
7 39
|
syl5d |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ Comp → ( ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o ) ) ) |
41 |
|
ordeleqon |
⊢ ( Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On ) ) |
42 |
|
unon |
⊢ ∪ On = On |
43 |
42
|
eqcomi |
⊢ On = ∪ On |
44 |
43
|
unieqi |
⊢ ∪ On = ∪ ∪ On |
45 |
|
unieq |
⊢ ( 𝐴 = On → ∪ 𝐴 = ∪ On ) |
46 |
45
|
unieqd |
⊢ ( 𝐴 = On → ∪ ∪ 𝐴 = ∪ ∪ On ) |
47 |
44 45 46
|
3eqtr4a |
⊢ ( 𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴 ) |
48 |
47
|
orim2i |
⊢ ( ( 𝐴 ∈ On ∨ 𝐴 = On ) → ( 𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴 ) ) |
49 |
41 48
|
sylbi |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ On ∨ ∪ 𝐴 = ∪ ∪ 𝐴 ) ) |
50 |
49
|
orcomd |
⊢ ( Ord 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 ∨ 𝐴 ∈ On ) ) |
51 |
50
|
ord |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ On ) ) |
52 |
|
unieq |
⊢ ( 𝐴 = ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴 ) |
53 |
52
|
con3i |
⊢ ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ¬ 𝐴 = ∪ 𝐴 ) |
54 |
34
|
ord |
⊢ ( Ord 𝐴 → ( ¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴 ) ) |
55 |
53 54
|
syl5 |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc ∪ 𝐴 ) ) |
56 |
|
orduniorsuc |
⊢ ( Ord ∪ 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴 ) ) |
57 |
1 56
|
syl |
⊢ ( Ord 𝐴 → ( ∪ 𝐴 = ∪ ∪ 𝐴 ∨ ∪ 𝐴 = suc ∪ ∪ 𝐴 ) ) |
58 |
57
|
ord |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → ∪ 𝐴 = suc ∪ ∪ 𝐴 ) ) |
59 |
|
suceq |
⊢ ( ∪ 𝐴 = suc ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 ) |
60 |
58 59
|
syl6 |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 ) ) |
61 |
|
eqtr |
⊢ ( ( 𝐴 = suc ∪ 𝐴 ∧ suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 ) → 𝐴 = suc suc ∪ ∪ 𝐴 ) |
62 |
61
|
ex |
⊢ ( 𝐴 = suc ∪ 𝐴 → ( suc ∪ 𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴 ) ) |
63 |
55 60 62
|
syl6c |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = suc suc ∪ ∪ 𝐴 ) ) |
64 |
|
onuni |
⊢ ( 𝐴 ∈ On → ∪ 𝐴 ∈ On ) |
65 |
|
onuni |
⊢ ( ∪ 𝐴 ∈ On → ∪ ∪ 𝐴 ∈ On ) |
66 |
|
onsucsuccmp |
⊢ ( ∪ ∪ 𝐴 ∈ On → suc suc ∪ ∪ 𝐴 ∈ Comp ) |
67 |
|
eleq1a |
⊢ ( suc suc ∪ ∪ 𝐴 ∈ Comp → ( 𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp ) ) |
68 |
64 65 66 67
|
4syl |
⊢ ( 𝐴 ∈ On → ( 𝐴 = suc suc ∪ ∪ 𝐴 → 𝐴 ∈ Comp ) ) |
69 |
51 63 68
|
syl6c |
⊢ ( Ord 𝐴 → ( ¬ ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 ∈ Comp ) ) |
70 |
|
id |
⊢ ( 𝐴 = 1o → 𝐴 = 1o ) |
71 |
70 16
|
eqtrdi |
⊢ ( 𝐴 = 1o → 𝐴 = { ∅ } ) |
72 |
|
0cmp |
⊢ { ∅ } ∈ Comp |
73 |
71 72
|
eqeltrdi |
⊢ ( 𝐴 = 1o → 𝐴 ∈ Comp ) |
74 |
73
|
a1i |
⊢ ( Ord 𝐴 → ( 𝐴 = 1o → 𝐴 ∈ Comp ) ) |
75 |
69 74
|
jad |
⊢ ( Ord 𝐴 → ( ( ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o ) → 𝐴 ∈ Comp ) ) |
76 |
40 75
|
impbid |
⊢ ( Ord 𝐴 → ( 𝐴 ∈ Comp ↔ ( ∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o ) ) ) |