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Theorem onsuct0 33863
Description: A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
Assertion
Ref Expression
onsuct0 (𝐴 ∈ On → suc 𝐴 ∈ Kol2)

Proof of Theorem onsuct0
Dummy variables 𝑜 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eloni 6179 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 df-ral 3135 . . . . . 6 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)))
3 ordelon 6193 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
4 ordelon 6193 . . . . . . . . . . 11 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
53, 4anim12dan 621 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6 ordsuc 7514 . . . . . . . . . . . 12 (Ord 𝐴 ↔ Ord suc 𝐴)
7 ordelon 6193 . . . . . . . . . . . . 13 ((Ord suc 𝐴𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
87ex 416 . . . . . . . . . . . 12 (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
96, 8sylbi 220 . . . . . . . . . . 11 (Ord 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
109adantr 484 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
11 notbi 322 . . . . . . . . . . . 12 ((𝑥𝑜𝑦𝑜) ↔ (¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜))
12 ontri1 6203 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥 ↔ ¬ 𝑥𝑜))
13 onsssuc 6256 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥𝑜 ∈ suc 𝑥))
1412, 13bitr3d 284 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
1514adantrr 716 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
16 ontri1 6203 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦 ↔ ¬ 𝑦𝑜))
17 onsssuc 6256 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦𝑜 ∈ suc 𝑦))
1816, 17bitr3d 284 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
1918adantrl 715 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
2015, 19bibi12d 349 . . . . . . . . . . . . 13 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2120ancoms 462 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2211, 21syl5bb 286 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2322biimpd 232 . . . . . . . . . 10 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
245, 10, 23syl6an 683 . . . . . . . . 9 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴 → ((𝑥𝑜𝑦𝑜) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))))
2524a2d 29 . . . . . . . 8 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → (𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))))
26 ordelss 6185 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
27 ordelord 6191 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
28 ordsucsssuc 7523 . . . . . . . . . . . . . . . 16 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
2928ancoms 462 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3027, 29syldan 594 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3126, 30mpbid 235 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥𝐴) → suc 𝑥 ⊆ suc 𝐴)
3231ssneld 3944 . . . . . . . . . . . 12 ((Ord 𝐴𝑥𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
3332adantrr 716 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34 ordelss 6185 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
35 ordelord 6191 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑦𝐴) → Ord 𝑦)
36 ordsucsssuc 7523 . . . . . . . . . . . . . . . 16 ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3736ancoms 462 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3835, 37syldan 594 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3934, 38mpbid 235 . . . . . . . . . . . . 13 ((Ord 𝐴𝑦𝐴) → suc 𝑦 ⊆ suc 𝐴)
4039ssneld 3944 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4140adantrl 715 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4233, 41jcad 516 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43 pm5.21 823 . . . . . . . . . 10 ((¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
4442, 43syl6 35 . . . . . . . . 9 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
45 idd 24 . . . . . . . . 9 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
4644, 45jad 190 . . . . . . . 8 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
4725, 46syld 47 . . . . . . 7 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
4847alimdv 1917 . . . . . 6 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
492, 48syl5bi 245 . . . . 5 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
50 dfcleq 2816 . . . . . . 7 (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
51 suc11 6272 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦𝑥 = 𝑦))
5250, 51bitr3id 288 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦) ↔ 𝑥 = 𝑦))
535, 52syl 17 . . . . 5 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦) ↔ 𝑥 = 𝑦))
5449, 53sylibd 242 . . . 4 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
5554ralrimivva 3181 . . 3 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
561, 55syl 17 . 2 (𝐴 ∈ On → ∀𝑥𝐴𝑦𝐴 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
57 onsuctopon 33856 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58 ist0-2 21947 . . 3 (suc 𝐴 ∈ (TopOn‘𝐴) → (suc 𝐴 ∈ Kol2 ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
5957, 58syl 17 . 2 (𝐴 ∈ On → (suc 𝐴 ∈ Kol2 ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
6056, 59mpbird 260 1 (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2114  wral 3130  wss 3908  Ord word 6168  Oncon0 6169  suc csuc 6171  cfv 6334  TopOnctopon 21513  Kol2ct0 21909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fv 6342  df-topgen 16708  df-top 21497  df-topon 21514  df-bases 21549  df-t0 21916
This theorem is referenced by:  ordtopt0  33864
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