Theorem onsucconi 30657

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 30653	. . 3 |- ( A e. On -> suc A e. Top )
3	1, 2	ax-mp 5	. 2 |- suc A e. Top
4		elin 3625	. . . 4 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) <-> ( x e. suc A /\ x e. ( Clsd ` suc A ) ) )
5		elsuci 4887	. . . . 5 |- ( x e. suc A -> ( x e. A \/ x = A ) )
6	1	onunisuci 4934	. . . . . . 7 |- U. suc A = A
7	6	eqcomi 2415	. . . . . 6 |- A = U. suc A
8	7	cldopn 19716	. . . . 5 |- ( x e. ( Clsd ` suc A ) -> ( A \ x ) e. suc A )
9	1	onsuci 6611	. . . . . . . . . 10 |- suc A e. On
10	9	oneli 4928	. . . . . . . . 9 |- ( ( A \ x ) e. suc A -> ( A \ x ) e. On )
11		elndif 3566	. . . . . . . . . . . 12 |- ( (/) e. x -> -. (/) e. ( A \ x ) )
12		on0eln0 4876	. . . . . . . . . . . . . 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 2621	. . . . . . . . . . . 12 |- ( ( A \ x ) e. On -> ( -. (/) e. ( A \ x ) -> ( A \ x ) = (/) ) )
15		ssdif0 3827	. . . . . . . . . . . . 13 |- ( A C_ x <-> ( A \ x ) = (/) )
16	1	onssneli 4930	. . . . . . . . . . . . 13 |- ( A C_ x -> -. x e. A )
17	15, 16	sylbir 213	. . . . . . . . . . . 12 |- ( ( A \ x ) = (/) -> -. x e. A )
18	11, 14, 17	syl56 32	. . . . . . . . . . 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 4928	. . . . . . . . . . . 12 |- ( x e. A -> x e. On )
21		on0eln0 4876	. . . . . . . . . . . . 13 |- ( x e. On -> ( (/) e. x <-> x =/= (/) ) )
22	21	biimprd 223	. . . . . . . . . . . 12 |- ( x e. On -> ( x =/= (/) -> (/) e. x ) )
23	20, 22	syl 17	. . . . . . . . . . 11 |- ( x e. A -> ( x =/= (/) -> (/) e. x ) )
24	23	necon1bd 2621	. . . . . . . . . 10 |- ( x e. A -> ( -. (/) e. x -> x = (/) ) )
25	19, 24	sylcom 27	. . . . . . . . 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 840	. . . . . . 7 |- ( ( A \ x ) e. suc A -> ( ( x e. A \/ x = A ) -> ( x = (/) \/ x = A ) ) )
28	27	impcom 428	. . . . . 6 |- ( ( ( x e. A \/ x = A ) /\ ( A \ x ) e. suc A ) -> ( x = (/) \/ x = A ) )
29		vex 3061	. . . . . . 7 |- x e. _V
30	29	elpr 3989	. . . . . 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 475	. . . 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 3445	. 2 |- ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A }
35	7	iscon2 20099	. 2 |- ( suc A e. Con <-> ( suc A e. Top /\ ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A } ) )
36	3, 34, 35	mpbir2an 921	1 |- suc A e. Con

Syntax hints:   -. wn 3   -> wi 4   \/ wo 366   /\ wa 367   = wceq 1405   e. wcel 1842   =/= wne 2598   \ cdif 3410   i^i cin 3412   C_ wss 3413   (/)c0 3737   {cpr 3973   U.cuni 4190   Oncon0 4821   suc csuc 4823   ` cfv 5525   Topctop 19578   Clsdccld 19701   Conccon 20096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533  df-topgen 14950  df-top 19583  df-bases 19585  df-cld 19704  df-con 20097

This theorem is referenced by:  onsuccon  30658