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Theorem ordcmp 31616
 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 6886 . . . 4 (Ord 𝐴 → Ord 𝐴)
2 unizlim 5761 . . . . . 6 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = ∅ ∨ Lim 𝐴)))
3 uni0b 4399 . . . . . . 7 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
43orbi1i 541 . . . . . 6 (( 𝐴 = ∅ ∨ Lim 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴))
52, 4syl6bb 275 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
65biimpd 218 . . . 4 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
71, 6syl 17 . . 3 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
8 sssn 4298 . . . . . . 7 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9 0ntop 20535 . . . . . . . . . . 11 ¬ ∅ ∈ Top
10 cmptop 21008 . . . . . . . . . . 11 (∅ ∈ Comp → ∅ ∈ Top)
119, 10mto 187 . . . . . . . . . 10 ¬ ∅ ∈ Comp
12 eleq1 2676 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
1311, 12mtbiri 316 . . . . . . . . 9 (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
1413pm2.21d 117 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
15 id 22 . . . . . . . . . 10 (𝐴 = {∅} → 𝐴 = {∅})
16 df1o2 7459 . . . . . . . . . 10 1𝑜 = {∅}
1715, 16syl6eqr 2662 . . . . . . . . 9 (𝐴 = {∅} → 𝐴 = 1𝑜)
1817a1d 25 . . . . . . . 8 (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
1914, 18jaoi 393 . . . . . . 7 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
208, 19sylbi 206 . . . . . 6 (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
2120a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1𝑜)))
22 ordtop 31605 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 𝐴))
2322biimpd 218 . . . . . . . . . 10 (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 𝐴))
2423necon2bd 2798 . . . . . . . . 9 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Top))
25 cmptop 21008 . . . . . . . . . 10 (𝐴 ∈ Comp → 𝐴 ∈ Top)
2625con3i 149 . . . . . . . . 9 𝐴 ∈ Top → ¬ 𝐴 ∈ Comp)
2724, 26syl6 34 . . . . . . . 8 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Comp))
2827a1dd 48 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
29 limsucncmp 31615 . . . . . . . . 9 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
30 eleq1 2676 . . . . . . . . . 10 (𝐴 = suc 𝐴 → (𝐴 ∈ Comp ↔ suc 𝐴 ∈ Comp))
3130notbid 307 . . . . . . . . 9 (𝐴 = suc 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc 𝐴 ∈ Comp))
3229, 31syl5ibr 235 . . . . . . . 8 (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
3332a1i 11 . . . . . . 7 (Ord 𝐴 → (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
34 orduniorsuc 6922 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
3528, 33, 34mpjaod 395 . . . . . 6 (Ord 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
36 pm2.21 119 . . . . . 6 𝐴 ∈ Comp → (𝐴 ∈ Comp → 𝐴 = 1𝑜))
3735, 36syl6 34 . . . . 5 (Ord 𝐴 → (Lim 𝐴 → (𝐴 ∈ Comp → 𝐴 = 1𝑜)))
3821, 37jaod 394 . . . 4 (Ord 𝐴 → ((𝐴 ⊆ {∅} ∨ Lim 𝐴) → (𝐴 ∈ Comp → 𝐴 = 1𝑜)))
3938com23 84 . . 3 (Ord 𝐴 → (𝐴 ∈ Comp → ((𝐴 ⊆ {∅} ∨ Lim 𝐴) → 𝐴 = 1𝑜)))
407, 39syl5d 71 . 2 (Ord 𝐴 → (𝐴 ∈ Comp → ( 𝐴 = 𝐴𝐴 = 1𝑜)))
41 ordeleqon 6880 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42 unon 6923 . . . . . . . . . . 11 On = On
4342eqcomi 2619 . . . . . . . . . 10 On = On
4443unieqi 4381 . . . . . . . . 9 On = On
45 unieq 4380 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4645unieqd 4382 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4744, 45, 463eqtr4a 2670 . . . . . . . 8 (𝐴 = On → 𝐴 = 𝐴)
4847orim2i 539 . . . . . . 7 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
4941, 48sylbi 206 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
5049orcomd 402 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴𝐴 ∈ On))
5150ord 391 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
52 unieq 4380 . . . . . . 7 (𝐴 = 𝐴 𝐴 = 𝐴)
5352con3i 149 . . . . . 6 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴)
5434ord 391 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
5553, 54syl5 33 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
56 orduniorsuc 6922 . . . . . . . 8 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
571, 56syl 17 . . . . . . 7 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
5857ord 391 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴 𝐴 = suc 𝐴))
59 suceq 5707 . . . . . 6 ( 𝐴 = suc 𝐴 → suc 𝐴 = suc suc 𝐴)
6058, 59syl6 34 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴 → suc 𝐴 = suc suc 𝐴))
61 eqtr 2629 . . . . . 6 ((𝐴 = suc 𝐴 ∧ suc 𝐴 = suc suc 𝐴) → 𝐴 = suc suc 𝐴)
6261ex 449 . . . . 5 (𝐴 = suc 𝐴 → (suc 𝐴 = suc suc 𝐴𝐴 = suc suc 𝐴))
6355, 60, 62syl6c 68 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc suc 𝐴))
64 onuni 6885 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
65 onuni 6885 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ On)
66 onsucsuccmp 31613 . . . . 5 ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
67 eleq1a 2683 . . . . 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 68 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ Comp))
70 id 22 . . . . . 6 (𝐴 = 1𝑜𝐴 = 1𝑜)
7170, 16syl6eq 2660 . . . . 5 (𝐴 = 1𝑜𝐴 = {∅})
72 0cmp 21007 . . . . 5 {∅} ∈ Comp
7371, 72syl6eqel 2696 . . . 4 (𝐴 = 1𝑜𝐴 ∈ Comp)
7473a1i 11 . . 3 (Ord 𝐴 → (𝐴 = 1𝑜𝐴 ∈ Comp))
7569, 74jad 173 . 2 (Ord 𝐴 → (( 𝐴 = 𝐴𝐴 = 1𝑜) → 𝐴 ∈ Comp))
7640, 75impbid 201 1 (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1𝑜)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  1𝑜c1o 7440  Topctop 20517  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000 This theorem is referenced by: (None)
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