Theorem onsuct0 34630

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 6276	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		df-ral 3069	. . . . . 6 ⊢ (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
3		ordelon 6290	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
4		ordelon 6290	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
5	3, 4	anim12dan 619	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
6		ordsuc 7661	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
7		ordelon 6290	. . . . . . . . . . . . 13 ⊢ ((Ord suc 𝐴 ∧ 𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
8	7	ex 413	. . . . . . . . . . . 12 ⊢ (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
9	6, 8	sylbi 216	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
10	9	adantr 481	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
11		notbi 319	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜))
12		ontri1 6300	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑜))
13		onsssuc 6353	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ 𝑜 ∈ suc 𝑥))
14	12, 13	bitr3d 280	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
15	14	adantrr 714	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
16		ontri1 6300	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑜))
17		onsssuc 6353	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ 𝑜 ∈ suc 𝑦))
18	16, 17	bitr3d 280	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
19	18	adantrl 713	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
20	15, 19	bibi12d 346	. . . . . . . . . . . . 13 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
21	20	ancoms 459	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
22	11, 21	syl5bb 283	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
23	22	biimpd 228	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
24	5, 10, 23	syl6an 681	. . . . . . . . 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 6282	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
27		ordelord 6288	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
28		ordsucsssuc 7670	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
29	28	ancoms 459	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
30	27, 29	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
31	26, 30	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc 𝐴)
32	31	ssneld 3923	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
33	32	adantrr 714	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34		ordelss 6282	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)
35		ordelord 6288	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → Ord 𝑦)
36		ordsucsssuc 7670	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
37	36	ancoms 459	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
38	35, 37	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
39	34, 38	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → suc 𝑦 ⊆ suc 𝐴)
40	39	ssneld 3923	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
41	40	adantrl 713	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
42	33, 41	jcad 513	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43		pm5.21 822	. . . . . . . . . 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 187	. . . . . . . 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 1919	. . . . . 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	syl5bi 241	. . . . 5 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
50		dfcleq 2731	. . . . . . 7 ⊢ (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))
51		suc11 6369	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦))
52	50, 51	bitr3id 285	. . . . . 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 238	. . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
55	54	ralrimivva 3123	. . 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 34623	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58		ist0-2 22495	. . 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 256	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  Ord word 6265  Oncon0 6266  suc csuc 6268  ‘cfv 6433  TopOnctopon 22059  Kol2ct0 22457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-t0 22464

This theorem is referenced by:  ordtopt0  34631