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Theorem onsuct0 32777
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 5876 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 df-ral 3066 . . . . . 6 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)))
3 ordelon 5890 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
4 ordelon 5890 . . . . . . . . . . 11 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
53, 4anim12dan 605 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6 ordsuc 7161 . . . . . . . . . . . 12 (Ord 𝐴 ↔ Ord suc 𝐴)
7 ordelon 5890 . . . . . . . . . . . . 13 ((Ord suc 𝐴𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
87ex 397 . . . . . . . . . . . 12 (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
96, 8sylbi 207 . . . . . . . . . . 11 (Ord 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
109adantr 466 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
11 notbi 308 . . . . . . . . . . . 12 ((𝑥𝑜𝑦𝑜) ↔ (¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜))
12 ontri1 5900 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥 ↔ ¬ 𝑥𝑜))
13 onsssuc 5956 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥𝑜 ∈ suc 𝑥))
1412, 13bitr3d 270 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
1514adantrr 696 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
16 ontri1 5900 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦 ↔ ¬ 𝑦𝑜))
17 onsssuc 5956 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦𝑜 ∈ suc 𝑦))
1816, 17bitr3d 270 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
1918adantrl 695 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
2015, 19bibi12d 334 . . . . . . . . . . . . 13 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2120ancoms 455 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2211, 21syl5bb 272 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2322biimpd 219 . . . . . . . . . 10 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
245, 10, 23syl6an 663 . . . . . . . . 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 5882 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
27 ordelord 5888 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
28 ordsucsssuc 7170 . . . . . . . . . . . . . . . 16 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
2928ancoms 455 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3027, 29syldan 579 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3126, 30mpbid 222 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥𝐴) → suc 𝑥 ⊆ suc 𝐴)
3231ssneld 3754 . . . . . . . . . . . 12 ((Ord 𝐴𝑥𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
3332adantrr 696 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34 ordelss 5882 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
35 ordelord 5888 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑦𝐴) → Ord 𝑦)
36 ordsucsssuc 7170 . . . . . . . . . . . . . . . 16 ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3736ancoms 455 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3835, 37syldan 579 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3934, 38mpbid 222 . . . . . . . . . . . . 13 ((Ord 𝐴𝑦𝐴) → suc 𝑦 ⊆ suc 𝐴)
4039ssneld 3754 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4140adantrl 695 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4233, 41jcad 502 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43 pm5.21 821 . . . . . . . . . 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 175 . . . . . . . 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 1997 . . . . . 6 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
492, 48syl5bi 232 . . . . 5 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
50 dfcleq 2765 . . . . . . 7 (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
51 suc11 5974 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦𝑥 = 𝑦))
5250, 51syl5bbr 274 . . . . . 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 229 . . . 4 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
5554ralrimivva 3120 . . 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 32770 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58 ist0-2 21369 . . 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 247 1 (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  wral 3061  wss 3723  Ord word 5865  Oncon0 5866  suc csuc 5868  cfv 6031  TopOnctopon 20935  Kol2ct0 21331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-t0 21338
This theorem is referenced by:  ordtopt0  32778
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