Theorem onsucconni 36439

Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)

Hypothesis

Ref	Expression
onsucconni.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onsucconni	⊢ suc 𝐴 ∈ Conn

Proof of Theorem onsucconni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucconni.1	. . 3 ⊢ 𝐴 ∈ On
2		onsuctop 36435	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
3	1, 2	ax-mp 5	. 2 ⊢ suc 𝐴 ∈ Top
4		elin 3966	. . . 4 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)))
5		elsuci 6450	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
6	1	onunisuci 6503	. . . . . . 7 ⊢ ∪ suc 𝐴 = 𝐴
7	6	eqcomi 2745	. . . . . 6 ⊢ 𝐴 = ∪ suc 𝐴
8	7	cldopn 23040	. . . . 5 ⊢ (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴 ∖ 𝑥) ∈ suc 𝐴)
9	1	onsuci 7860	. . . . . . . . . 10 ⊢ suc 𝐴 ∈ On
10	9	oneli 6497	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝐴 ∖ 𝑥) ∈ On)
11		elndif 4132	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴 ∖ 𝑥))
12		on0eln0 6439	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) ≠ ∅))
13	12	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∈ On → ((𝐴 ∖ 𝑥) ≠ ∅ → ∅ ∈ (𝐴 ∖ 𝑥)))
14	13	necon1bd 2957	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (¬ ∅ ∈ (𝐴 ∖ 𝑥) → (𝐴 ∖ 𝑥) = ∅))
15		ssdif0 4365	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
16	1	onssneli 6499	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 → ¬ 𝑥 ∈ 𝐴)
17	15, 16	sylbir 235	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴)
18	11, 14, 17	syl56 36	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥 ∈ 𝐴))
19	18	con2d 134	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (𝑥 ∈ 𝐴 → ¬ ∅ ∈ 𝑥))
20	1	oneli 6497	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
21		on0eln0 6439	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
22	21	biimprd 248	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
23	20, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
24	23	necon1bd 2957	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
25	19, 24	sylcom 30	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (𝑥 ∈ 𝐴 → 𝑥 = ∅))
26	10, 25	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 = ∅))
27	26	orim1d 967	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
28	27	impcom 407	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) ∧ (𝐴 ∖ 𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29		vex 3483	. . . . . . 7 ⊢ 𝑥 ∈ V
30	29	elpr 4649	. . . . . 6 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
31	28, 30	sylibr 234	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) ∧ (𝐴 ∖ 𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
32	5, 8, 31	syl2an 596	. . . 4 ⊢ ((𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
33	4, 32	sylbi 217	. . 3 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
34	33	ssriv 3986	. 2 ⊢ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
35	7	isconn2 23423	. 2 ⊢ (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
36	3, 34, 35	mpbir2an 711	1 ⊢ suc 𝐴 ∈ Conn

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ≠ wne 2939   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {cpr 4627  ∪ cuni 4906  Oncon0 6383  suc csuc 6385  ‘cfv 6560  Topctop 22900  Clsdccld 23025  Conncconn 23420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-topgen 17489  df-top 22901  df-bases 22954  df-cld 23028  df-conn 23421

This theorem is referenced by:  onsucconn  36440