Theorem onsuct0 36754

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.

Step	Hyp	Ref	Expression
1		eloni 6350	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		df-ral 3076	. . . . . 6 ⊢ (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
3		ordelon 6364	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
4		ordelon 6364	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
5	3, 4	anim12dan 628	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6		ordsuc 7788	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
7		ordelon 6364	. . . . . . . . . . . . 13 ⊢ ((Ord suc 𝐴 ∧ 𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
8	7	ex 416	. . . . . . . . . . . 12 ⊢ (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
9	6, 8	sylbi 219	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
10	9	adantr 484	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
11		notbi 321	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜))
12		ontri1 6374	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑜))
13		onsssuc 6432	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ 𝑜 ∈ suc 𝑥))
14	12, 13	bitr3d 283	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
15	14	adantrr 727	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
16		ontri1 6374	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑜))
17		onsssuc 6432	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ 𝑜 ∈ suc 𝑦))
18	16, 17	bitr3d 283	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
19	18	adantrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
20	15, 19	bibi12d 347	. . . . . . . . . . . . 13 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
21	20	ancoms 462	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
22	11, 21	bitrid 285	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
23	22	biimpd 231	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
24	5, 10, 23	syl6an 694	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑜 ∈ suc 𝐴 → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))))
25	24	a2d 29	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) → (𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))))
26		ordelss 6356	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
27		ordelord 6362	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
28		ordsucsssuc 7797	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
29	28	ancoms 462	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
30	27, 29	syldan 600	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
31	26, 30	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc 𝐴)
32	31	ssneld 3938	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
33	32	adantrr 727	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34		ordelss 6356	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)
35		ordelord 6362	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → Ord 𝑦)
36		ordsucsssuc 7797	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
37	36	ancoms 462	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
38	35, 37	syldan 600	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
39	34, 38	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → suc 𝑦 ⊆ suc 𝐴)
40	39	ssneld 3938	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
41	40	adantrl 726	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
42	33, 41	jcad 520	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43		pm5.21 834	. . . . . . . . . 10 ⊢ ((¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))
44	42, 43	syl6 35	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
45		idd 24	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
46	44, 45	jad 188	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
47	25, 46	syld 47	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
48	47	alimdv 1935	. . . . . 6 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) → ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
49	2, 48	biimtrid 244	. . . . 5 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
50		dfcleq 2754	. . . . . . 7 ⊢ (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))
51		suc11 6449	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦))
52	50, 51	bitr3id 287	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦) ↔ 𝑥 = 𝑦))
53	5, 52	syl 17	. . . . 5 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦) ↔ 𝑥 = 𝑦))
54	49, 53	sylibd 241	. . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
55	54	ralrimivva 3204	. . 3 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
56	1, 55	syl 17	. 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
57		onsuctopon 36747	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58		ist0-2 23382	. . 3 ⊢ (suc 𝐴 ∈ (TopOn‘𝐴) → (suc 𝐴 ∈ Kol2 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
59	57, 58	syl 17	. 2 ⊢ (𝐴 ∈ On → (suc 𝐴 ∈ Kol2 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
60	56, 59	mpbird 259	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3904  Ord word 6339  Oncon0 6340  suc csuc 6342  ‘cfv 6515  TopOnctopon 22948  Kol2ct0 23344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6471  df-fun 6517  df-fv 6523  df-topgen 17453  df-top 22932  df-topon 22949  df-bases 22984  df-t0 23351

This theorem is referenced by:  ordtopt0  36755