Theorem indiscld 20895

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 20806	. . . . 5 |- { (/) , A } e. Top
2		indisuni 20807	. . . . . 6 |- ( _I ` A ) = U. { (/) , A }
3	2	iscld 20831	. . . . 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 473	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x C_ ( _I ` A ) )
6		dfss4 3858	. . . . . 6 |- ( x C_ ( _I ` A ) <-> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
7	5, 6	sylib 208	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
8		simpr 477	. . . . . . 7 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , A } )
9		indislem 20804	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
10	8, 9	syl6eleqr 2712	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } )
11		elpri 4197	. . . . . 6 |- ( ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } -> ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) )
12		difeq2 3722	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ (/) ) )
13		dif0 3950	. . . . . . . . 9 |- ( ( _I ` A ) \ (/) ) = ( _I ` A )
14	12, 13	syl6eq 2672	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( _I ` A ) )
15		fvex 6201	. . . . . . . . . 10 |- ( _I ` A ) e. _V
16	15	prid2 4298	. . . . . . . . 9 |- ( _I ` A ) e. { (/) , ( _I ` A ) }
17	16, 9	eleqtri 2699	. . . . . . . 8 |- ( _I ` A ) e. { (/) , A }
18	14, 17	syl6eqel 2709	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
19		difeq2 3722	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ ( _I ` A ) ) )
20		difid 3948	. . . . . . . . 9 |- ( ( _I ` A ) \ ( _I ` A ) ) = (/)
21	19, 20	syl6eq 2672	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = (/) )
22		0ex 4790	. . . . . . . . 9 |- (/) e. _V
23	22	prid1 4297	. . . . . . . 8 |- (/) e. { (/) , A }
24	21, 23	syl6eqel 2709	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
25	18, 24	jaoi 394	. . . . . 6 |- ( ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
26	10, 11, 25	3syl 18	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
27	7, 26	eqeltrrd 2702	. . . 4 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x e. { (/) , A } )
28	4, 27	sylbi 207	. . 3 |- ( x e. ( Clsd ` { (/) , A } ) -> x e. { (/) , A } )
29	28	ssriv 3607	. 2 |- ( Clsd ` { (/) , A } ) C_ { (/) , A }
30		0cld 20842	. . . . 5 |- ( { (/) , A } e. Top -> (/) e. ( Clsd ` { (/) , A } ) )
31	1, 30	ax-mp 5	. . . 4 |- (/) e. ( Clsd ` { (/) , A } )
32	2	topcld 20839	. . . . 5 |- ( { (/) , A } e. Top -> ( _I ` A ) e. ( Clsd ` { (/) , A } ) )
33	1, 32	ax-mp 5	. . . 4 |- ( _I ` A ) e. ( Clsd ` { (/) , A } )
34		prssi 4353	. . . 4 |- ( ( (/) e. ( Clsd ` { (/) , A } ) /\ ( _I ` A ) e. ( Clsd ` { (/) , A } ) ) -> { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } ) )
35	31, 33, 34	mp2an 708	. . 3 |- { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } )
36	9, 35	eqsstr3i 3636	. 2 |- { (/) , A } C_ ( Clsd ` { (/) , A } )
37	29, 36	eqssi 3619	1 |- ( Clsd ` { (/) , A } ) = { (/) , A }

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   (/)c0 3915   {cpr 4179   _I cid 5023   ` cfv 5888   Topctop 20698   Clsdccld 20820

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716  df-cld 20823

This theorem is referenced by: (None)