Theorem indiscld 19570

Description: The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
indiscld	|- ( Clsd ` { (/) , A } ) = { (/) , A }

Proof of Theorem indiscld

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		indistop 19481	. . . . 5 |- { (/) , A } e. Top
2		indisuni 19482	. . . . . 6 |- ( _I ` A ) = U. { (/) , A }
3	2	iscld 19506	. . . . 5 |- ( { (/) , A } e. Top -> ( x e. ( Clsd ` { (/) , A } ) <-> ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) ) )
4	1, 3	ax-mp 5	. . . 4 |- ( x e. ( Clsd ` { (/) , A } ) <-> ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) )
5		simpl 457	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x C_ ( _I ` A ) )
6		dfss4 3717	. . . . . 6 |- ( x C_ ( _I ` A ) <-> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
7	5, 6	sylib 196	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
8		simpr 461	. . . . . . 7 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , A } )
9		indislem 19479	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
10	8, 9	syl6eleqr 2542	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } )
11		elpri 4034	. . . . . 6 |- ( ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } -> ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) )
12		difeq2 3601	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ (/) ) )
13		dif0 3884	. . . . . . . . 9 |- ( ( _I ` A ) \ (/) ) = ( _I ` A )
14	12, 13	syl6eq 2500	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( _I ` A ) )
15		fvex 5866	. . . . . . . . . 10 |- ( _I ` A ) e. _V
16	15	prid2 4124	. . . . . . . . 9 |- ( _I ` A ) e. { (/) , ( _I ` A ) }
17	16, 9	eleqtri 2529	. . . . . . . 8 |- ( _I ` A ) e. { (/) , A }
18	14, 17	syl6eqel 2539	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
19		difeq2 3601	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ ( _I ` A ) ) )
20		difid 3882	. . . . . . . . 9 |- ( ( _I ` A ) \ ( _I ` A ) ) = (/)
21	19, 20	syl6eq 2500	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = (/) )
22		0ex 4567	. . . . . . . . 9 |- (/) e. _V
23	22	prid1 4123	. . . . . . . 8 |- (/) e. { (/) , A }
24	21, 23	syl6eqel 2539	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
25	18, 24	jaoi 379	. . . . . 6 |- ( ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
26	10, 11, 25	3syl 20	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
27	7, 26	eqeltrrd 2532	. . . 4 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x e. { (/) , A } )
28	4, 27	sylbi 195	. . 3 |- ( x e. ( Clsd ` { (/) , A } ) -> x e. { (/) , A } )
29	28	ssriv 3493	. 2 |- ( Clsd ` { (/) , A } ) C_ { (/) , A }
30		0cld 19517	. . . . 5 |- ( { (/) , A } e. Top -> (/) e. ( Clsd ` { (/) , A } ) )
31	1, 30	ax-mp 5	. . . 4 |- (/) e. ( Clsd ` { (/) , A } )
32	2	topcld 19514	. . . . 5 |- ( { (/) , A } e. Top -> ( _I ` A ) e. ( Clsd ` { (/) , A } ) )
33	1, 32	ax-mp 5	. . . 4 |- ( _I ` A ) e. ( Clsd ` { (/) , A } )
34		prssi 4171	. . . 4 |- ( ( (/) e. ( Clsd ` { (/) , A } ) /\ ( _I ` A ) e. ( Clsd ` { (/) , A } ) ) -> { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } ) )
35	31, 33, 34	mp2an 672	. . 3 |- { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } )
36	9, 35	eqsstr3i 3520	. 2 |- { (/) , A } C_ ( Clsd ` { (/) , A } )
37	29, 36	eqssi 3505	1 |- ( Clsd ` { (/) , A } ) = { (/) , A }

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1383   e. wcel 1804   \ cdif 3458   C_ wss 3461   (/)c0 3770   {cpr 4016   _I cid 4780   ` cfv 5578   Topctop 19372   Clsdccld 19495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-top 19377  df-topon 19380  df-cld 19498

This theorem is referenced by: (None)