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Theorem ordcmp 32141
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 6956 . . . 4 (Ord 𝐴 → Ord 𝐴)
2 unizlim 5813 . . . . . 6 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = ∅ ∨ Lim 𝐴)))
3 uni0b 4436 . . . . . . 7 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
43orbi1i 542 . . . . . 6 (( 𝐴 = ∅ ∨ Lim 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴))
52, 4syl6bb 276 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
65biimpd 219 . . . 4 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
71, 6syl 17 . . 3 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
8 sssn 4333 . . . . . . 7 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9 0ntop 20650 . . . . . . . . . . 11 ¬ ∅ ∈ Top
10 cmptop 21138 . . . . . . . . . . 11 (∅ ∈ Comp → ∅ ∈ Top)
119, 10mto 188 . . . . . . . . . 10 ¬ ∅ ∈ Comp
12 eleq1 2686 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
1311, 12mtbiri 317 . . . . . . . . 9 (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
1413pm2.21d 118 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
15 id 22 . . . . . . . . . 10 (𝐴 = {∅} → 𝐴 = {∅})
16 df1o2 7532 . . . . . . . . . 10 1𝑜 = {∅}
1715, 16syl6eqr 2673 . . . . . . . . 9 (𝐴 = {∅} → 𝐴 = 1𝑜)
1817a1d 25 . . . . . . . 8 (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
1914, 18jaoi 394 . . . . . . 7 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
208, 19sylbi 207 . . . . . 6 (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
2120a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜)))
22 ordtop 32130 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 𝐴))
2322biimpd 219 . . . . . . . . . 10 (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 𝐴))
2423necon2bd 2806 . . . . . . . . 9 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Top))
25 cmptop 21138 . . . . . . . . . 10 (𝐴 ∈ Comp → 𝐴 ∈ Top)
2625con3i 150 . . . . . . . . 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 32140 . . . . . . . . 9 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
30 eleq1 2686 . . . . . . . . . 10 (𝐴 = suc 𝐴 → (𝐴 ∈ Comp ↔ suc 𝐴 ∈ Comp))
3130notbid 308 . . . . . . . . 9 (𝐴 = suc 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc 𝐴 ∈ Comp))
3229, 31syl5ibr 236 . . . . . . . 8 (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
3332a1i 11 . . . . . . 7 (Ord 𝐴 → (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
34 orduniorsuc 6992 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
3528, 33, 34mpjaod 396 . . . . . 6 (Ord 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
36 pm2.21 120 . . . . . 6 𝐴 ∈ Comp → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
3735, 36syl6 35 . . . . 5 (Ord 𝐴 → (Lim 𝐴 → (𝐴 ∈ Comp → 𝐴 = 1𝑜)))
3821, 37jaod 395 . . . 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 6950 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42 unon 6993 . . . . . . . . . . 11 On = On
4342eqcomi 2630 . . . . . . . . . 10 On = On
4443unieqi 4418 . . . . . . . . 9 On = On
45 unieq 4417 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4645unieqd 4419 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4744, 45, 463eqtr4a 2681 . . . . . . . 8 (𝐴 = On → 𝐴 = 𝐴)
4847orim2i 540 . . . . . . 7 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
4941, 48sylbi 207 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
5049orcomd 403 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴𝐴 ∈ On))
5150ord 392 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
52 unieq 4417 . . . . . . 7 (𝐴 = 𝐴 𝐴 = 𝐴)
5352con3i 150 . . . . . 6 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴)
5434ord 392 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
5553, 54syl5 34 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
56 orduniorsuc 6992 . . . . . . . 8 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
571, 56syl 17 . . . . . . 7 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
5857ord 392 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴 𝐴 = suc 𝐴))
59 suceq 5759 . . . . . 6 ( 𝐴 = suc 𝐴 → suc 𝐴 = suc suc 𝐴)
6058, 59syl6 35 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴 → suc 𝐴 = suc suc 𝐴))
61 eqtr 2640 . . . . . 6 ((𝐴 = suc 𝐴 ∧ suc 𝐴 = suc suc 𝐴) → 𝐴 = suc suc 𝐴)
6261ex 450 . . . . 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 6955 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
65 onuni 6955 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ On)
66 onsucsuccmp 32138 . . . . 5 ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
67 eleq1a 2693 . . . . 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, 16syl6eq 2671 . . . . 5 (𝐴 = 1𝑜𝐴 = {∅})
72 0cmp 21137 . . . . 5 {∅} ∈ Comp
7371, 72syl6eqel 2706 . . . 4 (𝐴 = 1𝑜𝐴 ∈ Comp)
7473a1i 11 . . 3 (Ord 𝐴 → (𝐴 = 1𝑜𝐴 ∈ Comp))
7569, 74jad 174 . 2 (Ord 𝐴 → (( 𝐴 = 𝐴𝐴 = 1𝑜) → 𝐴 ∈ Comp))
7640, 75impbid 202 1 (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1𝑜)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383   = wceq 1480  wcel 1987  wne 2790  wss 3560  c0 3897  {csn 4155   cuni 4409  Ord word 5691  Oncon0 5692  Lim wlim 5693  suc csuc 5694  1𝑜c1o 7513  Topctop 20638  Compccmp 21129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-1o 7520  df-er 7702  df-en 7916  df-fin 7919  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130
This theorem is referenced by: (None)
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