Theorem onsucconi 29479

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

Hypothesis

Ref	Expression
onsucconi.1	|- A e. On

Assertion

Ref	Expression
onsucconi	|- suc A e. Con

Proof of Theorem onsucconi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucconi.1	. . 3 |- A e. On
2		onsuctop 29475	. . 3 |- ( A e. On -> suc A e. Top )
3	1, 2	ax-mp 5	. 2 |- suc A e. Top
4		elin 3687	. . . 4 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) <-> ( x e. suc A /\ x e. ( Clsd ` suc A ) ) )
5		elsuci 4944	. . . . 5 |- ( x e. suc A -> ( x e. A \/ x = A ) )
6	1	onunisuci 4991	. . . . . . 7 |- U. suc A = A
7	6	eqcomi 2480	. . . . . 6 |- A = U. suc A
8	7	cldopn 19298	. . . . 5 |- ( x e. ( Clsd ` suc A ) -> ( A \ x ) e. suc A )
9	1	onsuci 6651	. . . . . . . . . 10 |- suc A e. On
10	9	oneli 4985	. . . . . . . . 9 |- ( ( A \ x ) e. suc A -> ( A \ x ) e. On )
11		elndif 3628	. . . . . . . . . . . 12 |- ( (/) e. x -> -. (/) e. ( A \ x ) )
12		on0eln0 4933	. . . . . . . . . . . . . 14 |- ( ( A \ x ) e. On -> ( (/) e. ( A \ x ) <-> ( A \ x ) =/= (/) ) )
13	12	biimprd 223	. . . . . . . . . . . . 13 |- ( ( A \ x ) e. On -> ( ( A \ x ) =/= (/) -> (/) e. ( A \ x ) ) )
14	13	necon1bd 2685	. . . . . . . . . . . 12 |- ( ( A \ x ) e. On -> ( -. (/) e. ( A \ x ) -> ( A \ x ) = (/) ) )
15		ssdif0 3885	. . . . . . . . . . . . 13 |- ( A C_ x <-> ( A \ x ) = (/) )
16	1	onssneli 4987	. . . . . . . . . . . . 13 |- ( A C_ x -> -. x e. A )
17	15, 16	sylbir 213	. . . . . . . . . . . 12 |- ( ( A \ x ) = (/) -> -. x e. A )
18	11, 14, 17	syl56 34	. . . . . . . . . . 11 |- ( ( A \ x ) e. On -> ( (/) e. x -> -. x e. A ) )
19	18	con2d 115	. . . . . . . . . 10 |- ( ( A \ x ) e. On -> ( x e. A -> -. (/) e. x ) )
20	1	oneli 4985	. . . . . . . . . . . 12 |- ( x e. A -> x e. On )
21		on0eln0 4933	. . . . . . . . . . . . 13 |- ( x e. On -> ( (/) e. x <-> x =/= (/) ) )
22	21	biimprd 223	. . . . . . . . . . . 12 |- ( x e. On -> ( x =/= (/) -> (/) e. x ) )
23	20, 22	syl 16	. . . . . . . . . . 11 |- ( x e. A -> ( x =/= (/) -> (/) e. x ) )
24	23	necon1bd 2685	. . . . . . . . . 10 |- ( x e. A -> ( -. (/) e. x -> x = (/) ) )
25	19, 24	sylcom 29	. . . . . . . . 9 |- ( ( A \ x ) e. On -> ( x e. A -> x = (/) ) )
26	10, 25	syl 16	. . . . . . . 8 |- ( ( A \ x ) e. suc A -> ( x e. A -> x = (/) ) )
27	26	orim1d 837	. . . . . . 7 |- ( ( A \ x ) e. suc A -> ( ( x e. A \/ x = A ) -> ( x = (/) \/ x = A ) ) )
28	27	impcom 430	. . . . . 6 |- ( ( ( x e. A \/ x = A ) /\ ( A \ x ) e. suc A ) -> ( x = (/) \/ x = A ) )
29		vex 3116	. . . . . . 7 |- x e. _V
30	29	elpr 4045	. . . . . 6 |- ( x e. { (/) , A } <-> ( x = (/) \/ x = A ) )
31	28, 30	sylibr 212	. . . . 5 |- ( ( ( x e. A \/ x = A ) /\ ( A \ x ) e. suc A ) -> x e. { (/) , A } )
32	5, 8, 31	syl2an 477	. . . 4 |- ( ( x e. suc A /\ x e. ( Clsd ` suc A ) ) -> x e. { (/) , A } )
33	4, 32	sylbi 195	. . 3 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) -> x e. { (/) , A } )
34	33	ssriv 3508	. 2 |- ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A }
35	7	iscon2 19681	. 2 |- ( suc A e. Con <-> ( suc A e. Top /\ ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A } ) )
36	3, 34, 35	mpbir2an 918	1 |- suc A e. Con

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785   {cpr 4029   U.cuni 4245   Oncon0 4878   suc csuc 4880   ` cfv 5586   Topctop 19161   Clsdccld 19283   Conccon 19678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-topgen 14695  df-top 19166  df-bases 19168  df-cld 19286  df-con 19679

This theorem is referenced by:  onsuccon  29480