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Theorem onsuct0 36424
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 6396 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 df-ral 3060 . . . . . 6 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)))
3 ordelon 6410 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
4 ordelon 6410 . . . . . . . . . . 11 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
53, 4anim12dan 619 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6 ordsuc 7833 . . . . . . . . . . . 12 (Ord 𝐴 ↔ Ord suc 𝐴)
7 ordelon 6410 . . . . . . . . . . . . 13 ((Ord suc 𝐴𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
87ex 412 . . . . . . . . . . . 12 (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
96, 8sylbi 217 . . . . . . . . . . 11 (Ord 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
109adantr 480 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
11 notbi 319 . . . . . . . . . . . 12 ((𝑥𝑜𝑦𝑜) ↔ (¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜))
12 ontri1 6420 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥 ↔ ¬ 𝑥𝑜))
13 onsssuc 6476 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥𝑜 ∈ suc 𝑥))
1412, 13bitr3d 281 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
1514adantrr 717 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
16 ontri1 6420 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦 ↔ ¬ 𝑦𝑜))
17 onsssuc 6476 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦𝑜 ∈ suc 𝑦))
1816, 17bitr3d 281 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
1918adantrl 716 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
2015, 19bibi12d 345 . . . . . . . . . . . . 13 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2120ancoms 458 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2211, 21bitrid 283 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2322biimpd 229 . . . . . . . . . 10 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
245, 10, 23syl6an 684 . . . . . . . . 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 6402 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
27 ordelord 6408 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
28 ordsucsssuc 7843 . . . . . . . . . . . . . . . 16 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
2928ancoms 458 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3027, 29syldan 591 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3126, 30mpbid 232 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥𝐴) → suc 𝑥 ⊆ suc 𝐴)
3231ssneld 3997 . . . . . . . . . . . 12 ((Ord 𝐴𝑥𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
3332adantrr 717 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34 ordelss 6402 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
35 ordelord 6408 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑦𝐴) → Ord 𝑦)
36 ordsucsssuc 7843 . . . . . . . . . . . . . . . 16 ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3736ancoms 458 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3835, 37syldan 591 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3934, 38mpbid 232 . . . . . . . . . . . . 13 ((Ord 𝐴𝑦𝐴) → suc 𝑦 ⊆ suc 𝐴)
4039ssneld 3997 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4140adantrl 716 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4233, 41jcad 512 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43 pm5.21 825 . . . . . . . . . 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 187 . . . . . . . 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 1914 . . . . . 6 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
492, 48biimtrid 242 . . . . 5 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
50 dfcleq 2728 . . . . . . 7 (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
51 suc11 6493 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦𝑥 = 𝑦))
5250, 51bitr3id 285 . . . . . 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 239 . . . 4 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
5554ralrimivva 3200 . . 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 36417 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58 ist0-2 23368 . . 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 257 1 (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  wss 3963  Ord word 6385  Oncon0 6386  suc csuc 6388  cfv 6563  TopOnctopon 22932  Kol2ct0 23330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-t0 23337
This theorem is referenced by:  ordtopt0  36425
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