Theorem onsucconni 36376

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 36372	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
3	1, 2	ax-mp 5	. 2 ⊢ suc 𝐴 ∈ Top
4		elin 3940	. . . 4 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)))
5		elsuci 6417	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
6	1	onunisuci 6470	. . . . . . 7 ⊢ ∪ suc 𝐴 = 𝐴
7	6	eqcomi 2743	. . . . . 6 ⊢ 𝐴 = ∪ suc 𝐴
8	7	cldopn 22954	. . . . 5 ⊢ (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴 ∖ 𝑥) ∈ suc 𝐴)
9	1	onsuci 7827	. . . . . . . . . 10 ⊢ suc 𝐴 ∈ On
10	9	oneli 6464	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝐴 ∖ 𝑥) ∈ On)
11		elndif 4106	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴 ∖ 𝑥))
12		on0eln0 6406	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) ≠ ∅))
13	12	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∈ On → ((𝐴 ∖ 𝑥) ≠ ∅ → ∅ ∈ (𝐴 ∖ 𝑥)))
14	13	necon1bd 2949	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (¬ ∅ ∈ (𝐴 ∖ 𝑥) → (𝐴 ∖ 𝑥) = ∅))
15		ssdif0 4339	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
16	1	onssneli 6466	. . . . . . . . . . . . 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 6464	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
21		on0eln0 6406	. . . . . . . . . . . . 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 2949	. . . . . . . . . 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 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
30	29	elpr 4623	. . . . . 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 3960	. 2 ⊢ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
35	7	isconn2 23337	. 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 2931   ∖ cdif 3921   ∩ cin 3923   ⊆ wss 3924  ∅c0 4306  {cpr 4601  ∪ cuni 4880  Oncon0 6349  suc csuc 6351  ‘cfv 6527  Topctop 22816  Clsdccld 22939  Conncconn 23334

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 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-ord 6352  df-on 6353  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-fv 6535  df-topgen 17442  df-top 22817  df-bases 22869  df-cld 22942  df-conn 23335

This theorem is referenced by:  onsucconn  36377