Theorem onsuct0 36429

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 6342	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		df-ral 3045	. . . . . 6 ⊢ (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∀𝑜(𝑜 ∈ suc 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
3		ordelon 6356	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
4		ordelon 6356	. . . . . . . . . . 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 7788	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
7		ordelon 6356	. . . . . . . . . . . . 13 ⊢ ((Ord suc 𝐴 ∧ 𝑜 ∈ suc 𝐴) → 𝑜 ∈ On)
8	7	ex 412	. . . . . . . . . . . 12 ⊢ (Ord suc 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
9	6, 8	sylbi 217	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝑜 ∈ suc 𝐴 → 𝑜 ∈ On))
10	9	adantr 480	. . . . . . . . . 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 6366	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑜))
13		onsssuc 6424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (𝑜 ⊆ 𝑥 ↔ 𝑜 ∈ suc 𝑥))
14	12, 13	bitr3d 281	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
15	14	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑥 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑥))
16		ontri1 6366	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑜))
17		onsssuc 6424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (𝑜 ⊆ 𝑦 ↔ 𝑜 ∈ suc 𝑦))
18	16, 17	bitr3d 281	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∈ On ∧ 𝑦 ∈ On) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
19	18	adantrl 716	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ∈ suc 𝑦))
20	15, 19	bibi12d 345	. . . . . . . . . . . . 13 ⊢ ((𝑜 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
21	20	ancoms 458	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
22	11, 21	bitrid 283	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
23	22	biimpd 229	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ 𝑜 ∈ On) → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
24	5, 10, 23	syl6an 684	. . . . . . . . 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 6348	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
27		ordelord 6354	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
28		ordsucsssuc 7798	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ suc 𝑥 ⊆ suc 𝐴))
29	28	ancoms 458	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc 𝐴)
32	31	ssneld 3948	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
33	32	adantrr 717	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥))
34		ordelss 6348	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)
35		ordelord 6354	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → Ord 𝑦)
36		ordsucsssuc 7798	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ Ord 𝐴) → (𝑦 ⊆ 𝐴 ↔ suc 𝑦 ⊆ suc 𝐴))
37	36	ancoms 458	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → suc 𝑦 ⊆ suc 𝐴)
40	39	ssneld 3948	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
41	40	adantrl 716	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦))
42	33, 41	jcad 512	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑜 ∈ suc 𝐴 → (¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦)))
43		pm5.21 824	. . . . . . . . . 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 1916	. . . . . 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 242	. . . . 5 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦)))
50		dfcleq 2722	. . . . . . 7 ⊢ (suc 𝑥 = suc 𝑦 ↔ ∀𝑜(𝑜 ∈ suc 𝑥 ↔ 𝑜 ∈ suc 𝑦))
51		suc11 6441	. . . . . . 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 239	. . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑜 ∈ suc 𝐴(𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
55	54	ralrimivva 3180	. . 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 36422	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
58		ist0-2 23231	. . 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 257	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  Ord word 6331  Oncon0 6332  suc csuc 6334  ‘cfv 6511  TopOnctopon 22797  Kol2ct0 23193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fv 6519  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-t0 23200

This theorem is referenced by:  ordtopt0  36430