Theorem onsucconni 33898

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 33894	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
3	1, 2	ax-mp 5	. 2 ⊢ suc 𝐴 ∈ Top
4		elin 3897	. . . 4 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)))
5		elsuci 6225	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
6	1	onunisuci 6272	. . . . . . 7 ⊢ ∪ suc 𝐴 = 𝐴
7	6	eqcomi 2807	. . . . . 6 ⊢ 𝐴 = ∪ suc 𝐴
8	7	cldopn 21636	. . . . 5 ⊢ (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴 ∖ 𝑥) ∈ suc 𝐴)
9	1	onsuci 7533	. . . . . . . . . 10 ⊢ suc 𝐴 ∈ On
10	9	oneli 6266	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝐴 ∖ 𝑥) ∈ On)
11		elndif 4056	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴 ∖ 𝑥))
12		on0eln0 6214	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) ≠ ∅))
13	12	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∈ On → ((𝐴 ∖ 𝑥) ≠ ∅ → ∅ ∈ (𝐴 ∖ 𝑥)))
14	13	necon1bd 3005	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (¬ ∅ ∈ (𝐴 ∖ 𝑥) → (𝐴 ∖ 𝑥) = ∅))
15		ssdif0 4277	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
16	1	onssneli 6268	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 → ¬ 𝑥 ∈ 𝐴)
17	15, 16	sylbir 238	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴)
18	11, 14, 17	syl56 36	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥 ∈ 𝐴))
19	18	con2d 136	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (𝑥 ∈ 𝐴 → ¬ ∅ ∈ 𝑥))
20	1	oneli 6266	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
21		on0eln0 6214	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
22	21	biimprd 251	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
23	20, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
24	23	necon1bd 3005	. . . . . . . . . 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 963	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
28	27	impcom 411	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) ∧ (𝐴 ∖ 𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
30	29	elpr 4548	. . . . . 6 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
31	28, 30	sylibr 237	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) ∧ (𝐴 ∖ 𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
32	5, 8, 31	syl2an 598	. . . 4 ⊢ ((𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
33	4, 32	sylbi 220	. . 3 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
34	33	ssriv 3919	. 2 ⊢ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
35	7	isconn2 22019	. 2 ⊢ (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
36	3, 34, 35	mpbir2an 710	1 ⊢ suc 𝐴 ∈ Conn

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {cpr 4527  ∪ cuni 4800  Oncon0 6159  suc csuc 6161  ‘cfv 6324  Topctop 21498  Clsdccld 21621  Conncconn 22016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-topgen 16709  df-top 21499  df-bases 21551  df-cld 21624  df-conn 22017

This theorem is referenced by:  onsucconn  33899