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 = 𝐴𝐴 = 1o ) ) )

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 = 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 = 𝐴𝐴 = 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 = 𝐴𝐴 = 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 = 𝐴𝐴 ∈ On ) )
51 50 ord ( Ord 𝐴 → ( ¬ 𝐴 = 𝐴𝐴 ∈ On ) )
52 unieq ( 𝐴 = 𝐴 𝐴 = 𝐴 )
53 52 con3i ( ¬ 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴 )
54 34 ord ( Ord 𝐴 → ( ¬ 𝐴 = 𝐴𝐴 = suc 𝐴 ) )
55 53 54 syl5 ( Ord 𝐴 → ( ¬ 𝐴 = 𝐴𝐴 = 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 𝐴𝐴 = suc suc 𝐴 ) )
63 55 60 62 syl6c ( Ord 𝐴 → ( ¬ 𝐴 = 𝐴𝐴 = 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 𝐴𝐴 ∈ Comp ) )
68 64 65 66 67 4syl ( 𝐴 ∈ On → ( 𝐴 = suc suc 𝐴𝐴 ∈ Comp ) )
69 51 63 68 syl6c ( Ord 𝐴 → ( ¬ 𝐴 = 𝐴𝐴 ∈ Comp ) )
70 id ( 𝐴 = 1o𝐴 = 1o )
71 70 16 eqtrdi ( 𝐴 = 1o𝐴 = { ∅ } )
72 0cmp { ∅ } ∈ Comp
73 71 72 eqeltrdi ( 𝐴 = 1o𝐴 ∈ Comp )
74 73 a1i ( Ord 𝐴 → ( 𝐴 = 1o𝐴 ∈ Comp ) )
75 69 74 jad ( Ord 𝐴 → ( ( 𝐴 = 𝐴𝐴 = 1o ) → 𝐴 ∈ Comp ) )
76 40 75 impbid ( Ord 𝐴 → ( 𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1o ) ) )