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Theorem onsuct0 33315
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 6039 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 df-ral 3093 . . . . . 6 (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)))
3 ordelon 6053 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
4 ordelon 6053 . . . . . . . . . . 11 ((Ord 𝐴𝑦𝐴) → 𝑦 ∈ On)
53, 4anim12dan 609 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6 ordsuc 7345 . . . . . . . . . . . 12 (Ord 𝐴 ↔ Ord suc 𝐴)
7 ordelon 6053 . . . . . . . . . . . . 13 ((Ord suc 𝐴𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
87ex 405 . . . . . . . . . . . 12 (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
96, 8sylbi 209 . . . . . . . . . . 11 (Ord 𝐴 → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
109adantr 473 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑜 ∈ suc 𝐴𝑜 ∈ On))
11 notbi 311 . . . . . . . . . . . 12 ((𝑥𝑜𝑦𝑜) ↔ (¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜))
12 ontri1 6063 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥 ↔ ¬ 𝑥𝑜))
13 onsssuc 6116 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜𝑥𝑜 ∈ suc 𝑥))
1412, 13bitr3d 273 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
1514adantrr 704 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥𝑜𝑜 ∈ suc 𝑥))
16 ontri1 6063 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦 ↔ ¬ 𝑦𝑜))
17 onsssuc 6116 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜𝑦𝑜 ∈ suc 𝑦))
1816, 17bitr3d 273 . . . . . . . . . . . . . . 15 ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
1918adantrl 703 . . . . . . . . . . . . . 14 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦𝑜𝑜 ∈ suc 𝑦))
2015, 19bibi12d 338 . . . . . . . . . . . . 13 ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2120ancoms 451 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2211, 21syl5bb 275 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) ↔ (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
2322biimpd 221 . . . . . . . . . 10 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥𝑜𝑦𝑜) → (𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
245, 10, 23syl6an 671 . . . . . . . . 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 6045 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
27 ordelord 6051 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
28 ordsucsssuc 7354 . . . . . . . . . . . . . . . 16 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
2928ancoms 451 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3027, 29syldan 582 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑥𝐴) → (𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
3126, 30mpbid 224 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥𝐴) → suc 𝑥 ⊆ suc 𝐴)
3231ssneld 3860 . . . . . . . . . . . 12 ((Ord 𝐴𝑥𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
3332adantrr 704 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34 ordelss 6045 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → 𝑦𝐴)
35 ordelord 6051 . . . . . . . . . . . . . . 15 ((Ord 𝐴𝑦𝐴) → Ord 𝑦)
36 ordsucsssuc 7354 . . . . . . . . . . . . . . . 16 ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3736ancoms 451 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3835, 37syldan 582 . . . . . . . . . . . . . 14 ((Ord 𝐴𝑦𝐴) → (𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
3934, 38mpbid 224 . . . . . . . . . . . . 13 ((Ord 𝐴𝑦𝐴) → suc 𝑦 ⊆ suc 𝐴)
4039ssneld 3860 . . . . . . . . . . . 12 ((Ord 𝐴𝑦𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4140adantrl 703 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
4233, 41jcad 505 . . . . . . . . . 10 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43 pm5.21 813 . . . . . . . . . 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 176 . . . . . . . 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 1875 . . . . . 6 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥𝑜𝑦𝑜)) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
492, 48syl5bi 234 . . . . 5 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦)))
50 dfcleq 2772 . . . . . . 7 (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦))
51 suc11 6132 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦𝑥 = 𝑦))
5250, 51syl5bbr 277 . . . . . 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 231 . . . 4 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
5554ralrimivva 3141 . . 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 33308 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58 ist0-2 21656 . . 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 249 1 (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1505   = wceq 1507  wcel 2050  wral 3088  wss 3829  Ord word 6028  Oncon0 6029  suc csuc 6031  cfv 6188  TopOnctopon 21222  Kol2ct0 21618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-ord 6032  df-on 6033  df-suc 6035  df-iota 6152  df-fun 6190  df-fv 6196  df-topgen 16573  df-top 21206  df-topon 21223  df-bases 21258  df-t0 21625
This theorem is referenced by:  ordtopt0  33316
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