Theorem indiscld 17110

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 17021	. . . . 5 |- { (/) , A } e. Top
2		indisuni 17022	. . . . . 6 |- ( _I ` A ) = U. { (/) , A }
3	2	iscld 17046	. . . . 5 |- ( { (/) , A } e. Top -> ( x e. ( Clsd ` { (/) , A } ) <-> ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) ) )
4	1, 3	ax-mp 8	. . . 4 |- ( x e. ( Clsd ` { (/) , A } ) <-> ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) )
5		simpl 444	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x C_ ( _I ` A ) )
6		dfss4 3535	. . . . . 6 |- ( x C_ ( _I ` A ) <-> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
7	5, 6	sylib 189	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = x )
8		simpr 448	. . . . . . 7 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , A } )
9		indislem 17019	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
10	8, 9	syl6eleqr 2495	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } )
11		elpri 3794	. . . . . 6 |- ( ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } -> ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) )
12		difeq2 3419	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ (/) ) )
13		dif0 3658	. . . . . . . . 9 |- ( ( _I ` A ) \ (/) ) = ( _I ` A )
14	12, 13	syl6eq 2452	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( _I ` A ) )
15		fvex 5701	. . . . . . . . . 10 |- ( _I ` A ) e. _V
16	15	prid2 3873	. . . . . . . . 9 |- ( _I ` A ) e. { (/) , ( _I ` A ) }
17	16, 9	eleqtri 2476	. . . . . . . 8 |- ( _I ` A ) e. { (/) , A }
18	14, 17	syl6eqel 2492	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
19		difeq2 3419	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ ( _I ` A ) ) )
20		difid 3656	. . . . . . . . 9 |- ( ( _I ` A ) \ ( _I ` A ) ) = (/)
21	19, 20	syl6eq 2452	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = (/) )
22		0ex 4299	. . . . . . . . 9 |- (/) e. _V
23	22	prid1 3872	. . . . . . . 8 |- (/) e. { (/) , A }
24	21, 23	syl6eqel 2492	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
25	18, 24	jaoi 369	. . . . . 6 |- ( ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
26	10, 11, 25	3syl 19	. . . . 5 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
27	7, 26	eqeltrrd 2479	. . . 4 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x e. { (/) , A } )
28	4, 27	sylbi 188	. . 3 |- ( x e. ( Clsd ` { (/) , A } ) -> x e. { (/) , A } )
29	28	ssriv 3312	. 2 |- ( Clsd ` { (/) , A } ) C_ { (/) , A }
30		0cld 17057	. . . . 5 |- ( { (/) , A } e. Top -> (/) e. ( Clsd ` { (/) , A } ) )
31	1, 30	ax-mp 8	. . . 4 |- (/) e. ( Clsd ` { (/) , A } )
32	2	topcld 17054	. . . . 5 |- ( { (/) , A } e. Top -> ( _I ` A ) e. ( Clsd ` { (/) , A } ) )
33	1, 32	ax-mp 8	. . . 4 |- ( _I ` A ) e. ( Clsd ` { (/) , A } )
34		prssi 3914	. . . 4 |- ( ( (/) e. ( Clsd ` { (/) , A } ) /\ ( _I ` A ) e. ( Clsd ` { (/) , A } ) ) -> { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } ) )
35	31, 33, 34	mp2an 654	. . 3 |- { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } )
36	9, 35	eqsstr3i 3339	. 2 |- { (/) , A } C_ ( Clsd ` { (/) , A } )
37	29, 36	eqssi 3324	1 |- ( Clsd ` { (/) , A } ) = { (/) , A }

Syntax hints:   <-> wb 177   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   \ cdif 3277   C_ wss 3280   (/)c0 3588   {cpr 3775   _I cid 4453   ` cfv 5413   Topctop 16913   Clsdccld 17035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-top 16918  df-topon 16921  df-cld 17038