Theorem onsucconi 28422

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 28418	. . 3 |- ( A e. On -> suc A e. Top )
3	1, 2	ax-mp 5	. 2 |- suc A e. Top
4		elin 3642	. . . 4 |- ( x e. ( suc A i^i ( Clsd ` suc A ) ) <-> ( x e. suc A /\ x e. ( Clsd ` suc A ) ) )
5		elsuci 4888	. . . . 5 |- ( x e. suc A -> ( x e. A \/ x = A ) )
6	1	onunisuci 4935	. . . . . . 7 |- U. suc A = A
7	6	eqcomi 2465	. . . . . 6 |- A = U. suc A
8	7	cldopn 18762	. . . . 5 |- ( x e. ( Clsd ` suc A ) -> ( A \ x ) e. suc A )
9	1	onsuci 6554	. . . . . . . . . 10 |- suc A e. On
10	9	oneli 4929	. . . . . . . . 9 |- ( ( A \ x ) e. suc A -> ( A \ x ) e. On )
11		elndif 3583	. . . . . . . . . . . 12 |- ( (/) e. x -> -. (/) e. ( A \ x ) )
12		on0eln0 4877	. . . . . . . . . . . . . 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 2667	. . . . . . . . . . . 12 |- ( ( A \ x ) e. On -> ( -. (/) e. ( A \ x ) -> ( A \ x ) = (/) ) )
15		ssdif0 3840	. . . . . . . . . . . . 13 |- ( A C_ x <-> ( A \ x ) = (/) )
16	1	onssneli 4931	. . . . . . . . . . . . 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 4929	. . . . . . . . . . . 12 |- ( x e. A -> x e. On )
21		on0eln0 4877	. . . . . . . . . . . . 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 2667	. . . . . . . . . 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 835	. . . . . . 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 3075	. . . . . . 7 |- x e. _V
30	29	elpr 3998	. . . . . 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 3463	. 2 |- ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A }
35	7	iscon2 19145	. 2 |- ( suc A e. Con <-> ( suc A e. Top /\ ( suc A i^i ( Clsd ` suc A ) ) C_ { (/) , A } ) )
36	3, 34, 35	mpbir2an 911	1 |- suc A e. Con

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2645   \ cdif 3428   i^i cin 3430   C_ wss 3431   (/)c0 3740   {cpr 3982   U.cuni 4194   Oncon0 4822   suc csuc 4824   ` cfv 5521   Topctop 18625   Clsdccld 18747   Conccon 19142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529  df-topgen 14496  df-top 18630  df-bases 18632  df-cld 18750  df-con 19143

This theorem is referenced by:  onsuccon  28423