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Theorem ordcmp 33682
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 7500 . . . 4 (Ord 𝐴 → Ord 𝐴)
2 unizlim 6304 . . . . . 6 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = ∅ ∨ Lim 𝐴)))
3 uni0b 4861 . . . . . . 7 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
43orbi1i 909 . . . . . 6 (( 𝐴 = ∅ ∨ Lim 𝐴) ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴))
52, 4syl6bb 288 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
65biimpd 230 . . . 4 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
71, 6syl 17 . . 3 (Ord 𝐴 → ( 𝐴 = 𝐴 → (𝐴 ⊆ {∅} ∨ Lim 𝐴)))
8 sssn 4757 . . . . . . 7 (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
9 0ntop 21432 . . . . . . . . . . 11 ¬ ∅ ∈ Top
10 cmptop 21922 . . . . . . . . . . 11 (∅ ∈ Comp → ∅ ∈ Top)
119, 10mto 198 . . . . . . . . . 10 ¬ ∅ ∈ Comp
12 eleq1 2904 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ∈ Comp ↔ ∅ ∈ Comp))
1311, 12mtbiri 328 . . . . . . . . 9 (𝐴 = ∅ → ¬ 𝐴 ∈ Comp)
1413pm2.21d 121 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ Comp → 𝐴 = 1o))
15 id 22 . . . . . . . . . 10 (𝐴 = {∅} → 𝐴 = {∅})
16 df1o2 8110 . . . . . . . . . 10 1o = {∅}
1715, 16syl6eqr 2878 . . . . . . . . 9 (𝐴 = {∅} → 𝐴 = 1o)
1817a1d 25 . . . . . . . 8 (𝐴 = {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
1914, 18jaoi 853 . . . . . . 7 ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ∈ Comp → 𝐴 = 1o))
208, 19sylbi 218 . . . . . 6 (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o))
2120a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ⊆ {∅} → (𝐴 ∈ Comp → 𝐴 = 1o)))
22 ordtop 33671 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∈ Top ↔ 𝐴 𝐴))
2322biimpd 230 . . . . . . . . . 10 (Ord 𝐴 → (𝐴 ∈ Top → 𝐴 𝐴))
2423necon2bd 3036 . . . . . . . . 9 (Ord 𝐴 → (𝐴 = 𝐴 → ¬ 𝐴 ∈ Top))
25 cmptop 21922 . . . . . . . . . 10 (𝐴 ∈ Comp → 𝐴 ∈ Top)
2625con3i 157 . . . . . . . . 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 33681 . . . . . . . . 9 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
30 eleq1 2904 . . . . . . . . . 10 (𝐴 = suc 𝐴 → (𝐴 ∈ Comp ↔ suc 𝐴 ∈ Comp))
3130notbid 319 . . . . . . . . 9 (𝐴 = suc 𝐴 → (¬ 𝐴 ∈ Comp ↔ ¬ suc 𝐴 ∈ Comp))
3229, 31syl5ibr 247 . . . . . . . 8 (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp))
3332a1i 11 . . . . . . 7 (Ord 𝐴 → (𝐴 = suc 𝐴 → (Lim 𝐴 → ¬ 𝐴 ∈ Comp)))
34 orduniorsuc 7536 . . . . . . 7 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
3528, 33, 34mpjaod 856 . . . . . 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 855 . . . 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 7494 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
42 unon 7537 . . . . . . . . . . 11 On = On
4342eqcomi 2834 . . . . . . . . . 10 On = On
4443unieqi 4845 . . . . . . . . 9 On = On
45 unieq 4844 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4645unieqd 4846 . . . . . . . . 9 (𝐴 = On → 𝐴 = On)
4744, 45, 463eqtr4a 2886 . . . . . . . 8 (𝐴 = On → 𝐴 = 𝐴)
4847orim2i 906 . . . . . . 7 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
4941, 48sylbi 218 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
5049orcomd 867 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴𝐴 ∈ On))
5150ord 860 . . . 4 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
52 unieq 4844 . . . . . . 7 (𝐴 = 𝐴 𝐴 = 𝐴)
5352con3i 157 . . . . . 6 𝐴 = 𝐴 → ¬ 𝐴 = 𝐴)
5434ord 860 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
5553, 54syl5 34 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
56 orduniorsuc 7536 . . . . . . . 8 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
571, 56syl 17 . . . . . . 7 (Ord 𝐴 → ( 𝐴 = 𝐴 𝐴 = suc 𝐴))
5857ord 860 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = 𝐴 𝐴 = suc 𝐴))
59 suceq 6253 . . . . . 6 ( 𝐴 = suc 𝐴 → suc 𝐴 = suc suc 𝐴)
6058, 59syl6 35 . . . . 5 (Ord 𝐴 → (¬ 𝐴 = 𝐴 → suc 𝐴 = suc suc 𝐴))
61 eqtr 2845 . . . . . 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 7499 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
65 onuni 7499 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ On)
66 onsucsuccmp 33679 . . . . 5 ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
67 eleq1a 2912 . . . . 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 2876 . . . . 5 (𝐴 = 1o𝐴 = {∅})
72 0cmp 21921 . . . . 5 {∅} ∈ Comp
7371, 72syl6eqel 2925 . . . 4 (𝐴 = 1o𝐴 ∈ Comp)
7473a1i 11 . . 3 (Ord 𝐴 → (𝐴 = 1o𝐴 ∈ Comp))
7569, 74jad 188 . 2 (Ord 𝐴 → (( 𝐴 = 𝐴𝐴 = 1o) → 𝐴 ∈ Comp))
7640, 75impbid 213 1 (Ord 𝐴 → (𝐴 ∈ Comp ↔ ( 𝐴 = 𝐴𝐴 = 1o)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 843   = wceq 1530  wcel 2107  wne 3020  wss 3939  c0 4294  {csn 4563   cuni 4836  Ord word 6187  Oncon0 6188  Lim wlim 6189  suc csuc 6190  1oc1o 8089  Topctop 21420  Compccmp 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-1o 8096  df-er 8282  df-en 8502  df-fin 8505  df-topgen 16710  df-top 21421  df-topon 21438  df-bases 21473  df-cmp 21914
This theorem is referenced by: (None)
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