Theorem onsucconni 36948

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 36944	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
3	1, 2	ax-mp 5	. 2 ⊢ suc 𝐴 ∈ Top
4		elin 3921	. . . 4 ⊢ (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴 ∧ 𝑥 ∈ (Clsd‘suc 𝐴)))
5		elsuci 6430	. . . . 5 ⊢ (𝑥 ∈ suc 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴))
6	1	onunisuci 6482	. . . . . . 7 ⊢ ∪ suc 𝐴 = 𝐴
7	6	eqcomi 2772	. . . . . 6 ⊢ 𝐴 = ∪ suc 𝐴
8	7	cldopn 23188	. . . . 5 ⊢ (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴 ∖ 𝑥) ∈ suc 𝐴)
9	1	onsuci 7831	. . . . . . . . . 10 ⊢ suc 𝐴 ∈ On
10	9	oneli 6476	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝐴 ∖ 𝑥) ∈ On)
11		elndif 4087	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴 ∖ 𝑥))
12		on0eln0 6418	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) ≠ ∅))
13	12	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∈ On → ((𝐴 ∖ 𝑥) ≠ ∅ → ∅ ∈ (𝐴 ∖ 𝑥)))
14	13	necon1bd 2976	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (¬ ∅ ∈ (𝐴 ∖ 𝑥) → (𝐴 ∖ 𝑥) = ∅))
15		ssdif0 4321	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
16	1	onssneli 6478	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 → ¬ 𝑥 ∈ 𝐴)
17	15, 16	sylbir 238	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴)
18	11, 14, 17	syl56 37	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥 ∈ 𝐴))
19	18	con2d 135	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (𝑥 ∈ 𝐴 → ¬ ∅ ∈ 𝑥))
20	1	oneli 6476	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
21		on0eln0 6418	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
22	21	biimprd 251	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
23	20, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
24	23	necon1bd 2976	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
25	19, 24	sylcom 31	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) ∈ On → (𝑥 ∈ 𝐴 → 𝑥 = ∅))
26	10, 25	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 = ∅))
27	26	orim1d 981	. . . . . . 7 ⊢ ((𝐴 ∖ 𝑥) ∈ suc 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
28	27	impcom 412	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐴) ∧ (𝐴 ∖ 𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
30	29	elpr 4614	. . . . . 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 607	. . . 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 3941	. 2 ⊢ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
35	7	isconn2 23571	. 2 ⊢ (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
36	3, 34, 35	mpbir2an 723	1 ⊢ suc 𝐴 ∈ Conn

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {cpr 4591  ∪ cuni 4872  Oncon0 6360  suc csuc 6362  ‘cfv 6536  Topctop 23050  Clsdccld 23173  Conncconn 23568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-topgen 17491  df-top 23051  df-bases 23103  df-cld 23176  df-conn 23569

This theorem is referenced by:  onsucconn  36949