Theorem onsucconni 32436

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

Hypothesis

Ref	Expression
onsucconni.1	|- A e. On

Assertion

Ref	Expression
onsucconni	|- suc A e. Conn

Proof of Theorem onsucconni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucconni.1	. . 3 |- A e. On
2		onsuctop 32432	. . 3 |- ( A e. On -> suc A e. Top )
3	1, 2	ax-mp 5	. 2 |- suc A e. Top
4		elin 3796	. . . 4 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) <-> ( x e. suc A /\ x e. ( Clsd ` suc A ) ) )
5		elsuci 5791	. . . . 5 |- ( x e. suc A -> ( x e. A \/ x = A ) )
6	1	onunisuci 5841	. . . . . . 7 |- U. suc A = A
7	6	eqcomi 2631	. . . . . 6 |- A = U. suc A
8	7	cldopn 20835	. . . . 5 |- ( x e. ( Clsd ` suc A ) -> ( A \ x ) e. suc A )
9	1	onsuci 7038	. . . . . . . . . 10 |- suc A e. On
10	9	oneli 5835	. . . . . . . . 9 |- ( ( A \ x ) e. suc A -> ( A \ x ) e. On )
11		elndif 3734	. . . . . . . . . . . 12 |- ( (/) e. x -> -. (/) e. ( A \ x ) )
12		on0eln0 5780	. . . . . . . . . . . . . 14 |- ( ( A \ x ) e. On -> ( (/) e. ( A \ x ) <-> ( A \ x ) =/= (/) ) )
13	12	biimprd 238	. . . . . . . . . . . . 13 |- ( ( A \ x ) e. On -> ( ( A \ x ) =/= (/) -> (/) e. ( A \ x ) ) )
14	13	necon1bd 2812	. . . . . . . . . . . 12 |- ( ( A \ x ) e. On -> ( -. (/) e. ( A \ x ) -> ( A \ x ) = (/) ) )
15		ssdif0 3942	. . . . . . . . . . . . 13 |- ( A C_ x <-> ( A \ x ) = (/) )
16	1	onssneli 5837	. . . . . . . . . . . . 13 |- ( A C_ x -> -. x e. A )
17	15, 16	sylbir 225	. . . . . . . . . . . 12 |- ( ( A \ x ) = (/) -> -. x e. A )
18	11, 14, 17	syl56 36	. . . . . . . . . . 11 |- ( ( A \ x ) e. On -> ( (/) e. x -> -. x e. A ) )
19	18	con2d 129	. . . . . . . . . 10 |- ( ( A \ x ) e. On -> ( x e. A -> -. (/) e. x ) )
20	1	oneli 5835	. . . . . . . . . . . 12 |- ( x e. A -> x e. On )
21		on0eln0 5780	. . . . . . . . . . . . 13 |- ( x e. On -> ( (/) e. x <-> x =/= (/) ) )
22	21	biimprd 238	. . . . . . . . . . . 12 |- ( x e. On -> ( x =/= (/) -> (/) e. x ) )
23	20, 22	syl 17	. . . . . . . . . . 11 |- ( x e. A -> ( x =/= (/) -> (/) e. x ) )
24	23	necon1bd 2812	. . . . . . . . . 10 |- ( x e. A -> ( -. (/) e. x -> x = (/) ) )
25	19, 24	sylcom 30	. . . . . . . . 9 |- ( ( A \ x ) e. On -> ( x e. A -> x = (/) ) )
26	10, 25	syl 17	. . . . . . . 8 |- ( ( A \ x ) e. suc A -> ( x e. A -> x = (/) ) )
27	26	orim1d 884	. . . . . . 7 |- ( ( A \ x ) e. suc A -> ( ( x e. A \/ x = A ) -> ( x = (/) \/ x = A ) ) )
28	27	impcom 446	. . . . . 6 |- ( ( ( x e. A \/ x = A ) /\ ( A \ x ) e. suc A ) -> ( x = (/) \/ x = A ) )
29		vex 3203	. . . . . . 7 |- x e. _V
30	29	elpr 4198	. . . . . 6 |- ( x e. { (/) , A } <-> ( x = (/) \/ x = A ) )
31	28, 30	sylibr 224	. . . . 5 |- ( ( ( x e. A \/ x = A ) /\ ( A \ x ) e. suc A ) -> x e. { (/) , A } )
32	5, 8, 31	syl2an 494	. . . 4 |- ( ( x e. suc A /\ x e. ( Clsd ` suc A ) ) -> x e. { (/) , A } )
33	4, 32	sylbi 207	. . 3 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) -> x e. { (/) , A } )
34	33	ssriv 3607	. 2 |- ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A }
35	7	isconn2 21217	. 2 |- ( suc A e. Conn <-> ( suc A e. Top /\ ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A } ) )
36	3, 34, 35	mpbir2an 955	1 |- suc A e. Conn

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   U.cuni 4436   Oncon0 5723   suc csuc 5725   ` cfv 5888   Topctop 20698   Clsdccld 20820  Conncconn 21214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-topgen 16104  df-top 20699  df-bases 20750  df-cld 20823  df-conn 21215

This theorem is referenced by:  onsucconn  32437