Theorem onsucconni 36398

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 36394	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
3	1, 2	ax-mp 5	. 2 ⊢ suc 𝐴 ∈ Top
4		elin 3927	. . . 4 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)))
5		elsuci 6389	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
6	1	onunisuci 6442	. . . . . . 7 ⊢ ∪ suc 𝐴 = 𝐴
7	6	eqcomi 2738	. . . . . 6 ⊢ 𝐴 = ∪ suc 𝐴
8	7	cldopn 22894	. . . . 5 ⊢ (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴 ∖ 𝑥) ∈ suc 𝐴)
9	1	onsuci 7794	. . . . . . . . . 10 ⊢ suc 𝐴 ∈ On
10	9	oneli 6436	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝐴 ∖ 𝑥) ∈ On)
11		elndif 4092	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴 ∖ 𝑥))
12		on0eln0 6377	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) ≠ ∅))
13	12	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∈ On → ((𝐴 ∖ 𝑥) ≠ ∅ → ∅ ∈ (𝐴 ∖ 𝑥)))
14	13	necon1bd 2943	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (¬ ∅ ∈ (𝐴 ∖ 𝑥) → (𝐴 ∖ 𝑥) = ∅))
15		ssdif0 4325	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
16	1	onssneli 6438	. . . . . . . . . . . . 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 6436	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
21		on0eln0 6377	. . . . . . . . . . . . 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 2943	. . . . . . . . . 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 3448	. . . . . . 7 ⊢ 𝑥 ∈ V
30	29	elpr 4610	. . . . . 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 3947	. 2 ⊢ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
35	7	isconn2 23277	. 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 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {cpr 4587  ∪ cuni 4867  Oncon0 6320  suc csuc 6322  ‘cfv 6499  Topctop 22756  Clsdccld 22879  Conncconn 23274

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-topgen 17382  df-top 22757  df-bases 22809  df-cld 22882  df-conn 23275

This theorem is referenced by:  onsucconn  36399