Metamath Proof Explorer


Theorem ordcmp

Description: An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1o . (Contributed by Chen-Pang He, 1-Nov-2015)

Ref Expression
Assertion ordcmp Ord A A Comp A = A A = 1 𝑜

Proof

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 = 1 𝑜
15 id A = A =
16 df1o2 1 𝑜 =
17 15 16 eqtr4di A = A = 1 𝑜
18 17 a1d A = A Comp A = 1 𝑜
19 14 18 jaoi A = A = A Comp A = 1 𝑜
20 8 19 sylbi A A Comp A = 1 𝑜
21 20 a1i Ord A A A Comp A = 1 𝑜
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 = 1 𝑜
37 35 36 syl6 Ord A Lim A A Comp A = 1 𝑜
38 21 37 jaod Ord A A Lim A A Comp A = 1 𝑜
39 38 com23 Ord A A Comp A Lim A A = 1 𝑜
40 7 39 syl5d Ord A A Comp A = A A = 1 𝑜
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 = 1 𝑜 A = 1 𝑜
71 70 16 eqtrdi A = 1 𝑜 A =
72 0cmp Comp
73 71 72 eqeltrdi A = 1 𝑜 A Comp
74 73 a1i Ord A A = 1 𝑜 A Comp
75 69 74 jad Ord A A = A A = 1 𝑜 A Comp
76 40 75 impbid Ord A A Comp A = A A = 1 𝑜