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Theorem ordcmp 36448
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 7809 . . . 4 (Ord 𝐴 → Ord 𝐴)
2 unizlim 6507 . . . . . 6 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = ∅ ∨ Lim 𝐴)))
3 uni0b 4933 . . . . . . 7 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
43orbi1i 914 . . . . . 6 (( 𝐴 = ∅ ∨ Lim 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴))
52, 4bitrdi 287 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
65biimpd 229 . . . 4 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
71, 6syl 17 . . 3 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
8 sssn 4826 . . . . . . 7 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9 0ntop 22911 . . . . . . . . . . 11 ¬ ∅ ∈ Top
10 cmptop 23403 . . . . . . . . . . 11 (∅ ∈ Comp → ∅ ∈ Top)
119, 10mto 197 . . . . . . . . . 10 ¬ ∅ ∈ Comp
12 eleq1 2829 . . . . . . . . . 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 8513 . . . . . . . . . 10 1o = {∅}
1715, 16eqtr4di 2795 . . . . . . . . 9 (𝐴 = {∅} → 𝐴 = 1o)
1817a1d 25 . . . . . . . 8 (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
1914, 18jaoi 858 . . . . . . 7 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
208, 19sylbi 217 . . . . . 6 (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
2120a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22 ordtop 36437 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 𝐴))
2322biimpd 229 . . . . . . . . . 10 (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 𝐴))
2423necon2bd 2956 . . . . . . . . 9 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Top))
25 cmptop 23403 . . . . . . . . . 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 36447 . . . . . . . . 9 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
30 eleq1 2829 . . . . . . . . . 10 (𝐴 = suc 𝐴 → (𝐴 ∈ Comp ↔ suc 𝐴 ∈ Comp))
3130notbid 318 . . . . . . . . 9 (𝐴 = suc 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc 𝐴 ∈ Comp))
3229, 31imbitrrid 246 . . . . . . . 8 (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
3332a1i 11 . . . . . . 7 (Ord 𝐴 → (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
34 orduniorsuc 7850 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
3528, 33, 34mpjaod 861 . . . . . 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 860 . . . 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 7802 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42 unon 7851 . . . . . . . . . . 11 On = On
4342eqcomi 2746 . . . . . . . . . 10 On = On
4443unieqi 4919 . . . . . . . . 9 On = On
45 unieq 4918 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4645unieqd 4920 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4744, 45, 463eqtr4a 2803 . . . . . . . 8 (𝐴 = On → 𝐴 = 𝐴)
4847orim2i 911 . . . . . . 7 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
4941, 48sylbi 217 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
5049orcomd 872 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴𝐴 ∈ On))
5150ord 865 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
52 unieq 4918 . . . . . . 7 (𝐴 = 𝐴 𝐴 = 𝐴)
5352con3i 154 . . . . . 6 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴)
5434ord 865 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
5553, 54syl5 34 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
56 orduniorsuc 7850 . . . . . . . 8 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
571, 56syl 17 . . . . . . 7 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
5857ord 865 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴 𝐴 = suc 𝐴))
59 suceq 6450 . . . . . 6 ( 𝐴 = suc 𝐴 → suc 𝐴 = suc suc 𝐴)
6058, 59syl6 35 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴 → suc 𝐴 = suc suc 𝐴))
61 eqtr 2760 . . . . . 6 ((𝐴 = suc 𝐴 ∧ suc 𝐴 = suc suc 𝐴) → 𝐴 = suc suc 𝐴)
6261ex 412 . . . . 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 7808 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
65 onuni 7808 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ On)
66 onsucsuccmp 36445 . . . . 5 ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
67 eleq1a 2836 . . . . 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 2793 . . . . 5 (𝐴 = 1o𝐴 = {∅})
72 0cmp 23402 . . . . 5 {∅} ∈ Comp
7371, 72eqeltrdi 2849 . . . 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 212 1 (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1o)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  wne 2940  wss 3951  c0 4333  {csn 4626   cuni 4907  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  1oc1o 8499  Topctop 22899  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395
This theorem is referenced by: (None)
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