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Theorem onsuct0 32082
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 5692 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 df-ral 2912 . . . . . 6 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)))
3 ordelon 5706 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
4 ordelon 5706 . . . . . . . . . . 11 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
53, 4anim12dan 881 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6 ordsuc 6961 . . . . . . . . . . . 12 (Ord 𝐴 ↔ Ord suc 𝐴)
7 ordelon 5706 . . . . . . . . . . . . 13 ((Ord suc 𝐴𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
87ex 450 . . . . . . . . . . . 12 (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
96, 8sylbi 207 . . . . . . . . . . 11 (Ord 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
109adantr 481 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
11 notbi 309 . . . . . . . . . . . 12 ((𝑥𝑜𝑦𝑜) ↔ (¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜))
12 ontri1 5716 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥 ↔ ¬ 𝑥𝑜))
13 onsssuc 5772 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥𝑜 ∈ suc 𝑥))
1412, 13bitr3d 270 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
1514adantrr 752 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
16 ontri1 5716 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦 ↔ ¬ 𝑦𝑜))
17 onsssuc 5772 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦𝑜 ∈ suc 𝑦))
1816, 17bitr3d 270 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
1918adantrl 751 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
2015, 19bibi12d 335 . . . . . . . . . . . . 13 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2120ancoms 469 . . . . . . . . . . . 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 567 . . . . . . . . 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 5698 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
27 ordelord 5704 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
28 ordsucsssuc 6970 . . . . . . . . . . . . . . . 16 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
2928ancoms 469 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3027, 29syldan 487 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3126, 30mpbid 222 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥𝐴) → suc 𝑥 ⊆ suc 𝐴)
3231ssneld 3585 . . . . . . . . . . . 12 ((Ord 𝐴𝑥𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
3332adantrr 752 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34 ordelss 5698 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
35 ordelord 5704 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑦𝐴) → Ord 𝑦)
36 ordsucsssuc 6970 . . . . . . . . . . . . . . . 16 ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3736ancoms 469 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3835, 37syldan 487 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3934, 38mpbid 222 . . . . . . . . . . . . 13 ((Ord 𝐴𝑦𝐴) → suc 𝑦 ⊆ suc 𝐴)
4039ssneld 3585 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4140adantrl 751 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4233, 41jcad 555 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43 pm5.21 902 . . . . . . . . . 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 174 . . . . . . . 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 1842 . . . . . 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 2615 . . . . . . 7 (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
51 suc11 5790 . . . . . . 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 2965 . . 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 32075 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58 ist0-2 21058 . . 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 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  wss 3555  Ord word 5681  Oncon0 5682  suc csuc 5684  cfv 5847  TopOnctopon 20618  Kol2ct0 21020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-t0 21027
This theorem is referenced by:  ordtopt0  32083
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