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Theorem ordcmp 34636
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.)
Assertion
Ref Expression
ordcmp (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1o)))

Proof of Theorem ordcmp
StepHypRef Expression
1 orduni 7639 . . . 4 (Ord 𝐴 → Ord 𝐴)
2 unizlim 6383 . . . . . 6 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = ∅ ∨ Lim 𝐴)))
3 uni0b 4867 . . . . . . 7 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
43orbi1i 911 . . . . . 6 (( 𝐴 = ∅ ∨ Lim 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴))
52, 4bitrdi 287 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
65biimpd 228 . . . 4 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
71, 6syl 17 . . 3 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
8 sssn 4759 . . . . . . 7 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9 0ntop 22054 . . . . . . . . . . 11 ¬ ∅ ∈ Top
10 cmptop 22546 . . . . . . . . . . 11 (∅ ∈ Comp → ∅ ∈ Top)
119, 10mto 196 . . . . . . . . . 10 ¬ ∅ ∈ Comp
12 eleq1 2826 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
1311, 12mtbiri 327 . . . . . . . . 9 (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
1413pm2.21d 121 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1o))
15 id 22 . . . . . . . . . 10 (𝐴 = {∅} → 𝐴 = {∅})
16 df1o2 8304 . . . . . . . . . 10 1o = {∅}
1715, 16eqtr4di 2796 . . . . . . . . 9 (𝐴 = {∅} → 𝐴 = 1o)
1817a1d 25 . . . . . . . 8 (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
1914, 18jaoi 854 . . . . . . 7 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
208, 19sylbi 216 . . . . . 6 (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
2120a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22 ordtop 34625 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 𝐴))
2322biimpd 228 . . . . . . . . . 10 (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 𝐴))
2423necon2bd 2959 . . . . . . . . 9 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Top))
25 cmptop 22546 . . . . . . . . . 10 (𝐴 ∈ Comp → 𝐴 ∈ Top)
2625con3i 154 . . . . . . . . 9 𝐴 ∈ Top → ¬ 𝐴 ∈ Comp)
2724, 26syl6 35 . . . . . . . 8 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Comp))
2827a1dd 50 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
29 limsucncmp 34635 . . . . . . . . 9 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
30 eleq1 2826 . . . . . . . . . 10 (𝐴 = suc 𝐴 → (𝐴 ∈ Comp ↔ suc 𝐴 ∈ Comp))
3130notbid 318 . . . . . . . . 9 (𝐴 = suc 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc 𝐴 ∈ Comp))
3229, 31syl5ibr 245 . . . . . . . 8 (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
3332a1i 11 . . . . . . 7 (Ord 𝐴 → (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
34 orduniorsuc 7677 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
3528, 33, 34mpjaod 857 . . . . . 6 (Ord 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
36 pm2.21 123 . . . . . 6 𝐴 ∈ Comp → (𝐴 ∈ Comp → 𝐴 = 1o))
3735, 36syl6 35 . . . . 5 (Ord 𝐴 → (Lim 𝐴 → (𝐴 ∈ Comp → 𝐴 = 1o)))
3821, 37jaod 856 . . . 4 (Ord 𝐴 → ((𝐴 ⊆ {∅} ∨ Lim 𝐴) → (𝐴 ∈ Comp → 𝐴 = 1o)))
3938com23 86 . . 3 (Ord 𝐴 → (𝐴 ∈ Comp → ((𝐴 ⊆ {∅} ∨ Lim 𝐴) → 𝐴 = 1o)))
407, 39syl5d 73 . 2 (Ord 𝐴 → (𝐴 ∈ Comp → ( 𝐴 = 𝐴𝐴 = 1o)))
41 ordeleqon 7632 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42 unon 7678 . . . . . . . . . . 11 On = On
4342eqcomi 2747 . . . . . . . . . 10 On = On
4443unieqi 4852 . . . . . . . . 9 On = On
45 unieq 4850 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4645unieqd 4853 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4744, 45, 463eqtr4a 2804 . . . . . . . 8 (𝐴 = On → 𝐴 = 𝐴)
4847orim2i 908 . . . . . . 7 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
4941, 48sylbi 216 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
5049orcomd 868 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴𝐴 ∈ On))
5150ord 861 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
52 unieq 4850 . . . . . . 7 (𝐴 = 𝐴 𝐴 = 𝐴)
5352con3i 154 . . . . . 6 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴)
5434ord 861 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
5553, 54syl5 34 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
56 orduniorsuc 7677 . . . . . . . 8 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
571, 56syl 17 . . . . . . 7 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
5857ord 861 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴 𝐴 = suc 𝐴))
59 suceq 6331 . . . . . 6 ( 𝐴 = suc 𝐴 → suc 𝐴 = suc suc 𝐴)
6058, 59syl6 35 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴 → suc 𝐴 = suc suc 𝐴))
61 eqtr 2761 . . . . . 6 ((𝐴 = suc 𝐴 ∧ suc 𝐴 = suc suc 𝐴) → 𝐴 = suc suc 𝐴)
6261ex 413 . . . . 5 (𝐴 = suc 𝐴 → (suc 𝐴 = suc suc 𝐴𝐴 = suc suc 𝐴))
6355, 60, 62syl6c 70 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc suc 𝐴))
64 onuni 7638 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
65 onuni 7638 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ On)
66 onsucsuccmp 34633 . . . . 5 ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
67 eleq1a 2834 . . . . 5 (suc suc 𝐴 ∈ Comp → (𝐴 = suc suc 𝐴𝐴 ∈ Comp))
6864, 65, 66, 674syl 19 . . . 4 (𝐴 ∈ On → (𝐴 = suc suc 𝐴𝐴 ∈ Comp))
6951, 63, 68syl6c 70 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ Comp))
70 id 22 . . . . . 6 (𝐴 = 1o𝐴 = 1o)
7170, 16eqtrdi 2794 . . . . 5 (𝐴 = 1o𝐴 = {∅})
72 0cmp 22545 . . . . 5 {∅} ∈ Comp
7371, 72eqeltrdi 2847 . . . 4 (𝐴 = 1o𝐴 ∈ Comp)
7473a1i 11 . . 3 (Ord 𝐴 → (𝐴 = 1o𝐴 ∈ Comp))
7569, 74jad 187 . 2 (Ord 𝐴 → (( 𝐴 = 𝐴𝐴 = 1o) → 𝐴 ∈ Comp))
7640, 75impbid 211 1 (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1o)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 844   = wceq 1539  wcel 2106  wne 2943  wss 3887  c0 4256  {csn 4561   cuni 4839  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  1oc1o 8290  Topctop 22042  Compccmp 22537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538
This theorem is referenced by: (None)
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