Theorem indiscld 19355

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 19266	. . . . 5 |- { (/) , A } e. Top
2		indisuni 19267	. . . . . 6 |- ( _I ` A ) = U. { (/) , A }
3	2	iscld 19291	. . . . 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 3732	. . . . . 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 19264	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
10	8, 9	syl6eleqr 2566	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } )
11		elpri 4047	. . . . . 6 |- ( ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } -> ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) )
12		difeq2 3616	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ (/) ) )
13		dif0 3897	. . . . . . . . 9 |- ( ( _I ` A ) \ (/) ) = ( _I ` A )
14	12, 13	syl6eq 2524	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( _I ` A ) )
15		fvex 5874	. . . . . . . . . 10 |- ( _I ` A ) e. _V
16	15	prid2 4136	. . . . . . . . 9 |- ( _I ` A ) e. { (/) , ( _I ` A ) }
17	16, 9	eleqtri 2553	. . . . . . . 8 |- ( _I ` A ) e. { (/) , A }
18	14, 17	syl6eqel 2563	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
19		difeq2 3616	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ ( _I ` A ) ) )
20		difid 3895	. . . . . . . . 9 |- ( ( _I ` A ) \ ( _I ` A ) ) = (/)
21	19, 20	syl6eq 2524	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = (/) )
22		0ex 4577	. . . . . . . . 9 |- (/) e. _V
23	22	prid1 4135	. . . . . . . 8 |- (/) e. { (/) , A }
24	21, 23	syl6eqel 2563	. . . . . . 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 2556	. . . 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 3508	. 2 |- ( Clsd ` { (/) , A } ) C_ { (/) , A }
30		0cld 19302	. . . . 5 |- ( { (/) , A } e. Top -> (/) e. ( Clsd ` { (/) , A } ) )
31	1, 30	ax-mp 5	. . . 4 |- (/) e. ( Clsd ` { (/) , A } )
32	2	topcld 19299	. . . . 5 |- ( { (/) , A } e. Top -> ( _I ` A ) e. ( Clsd ` { (/) , A } ) )
33	1, 32	ax-mp 5	. . . 4 |- ( _I ` A ) e. ( Clsd ` { (/) , A } )
34		prssi 4183	. . . 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 3535	. 2 |- { (/) , A } C_ ( Clsd ` { (/) , A } )
37	29, 36	eqssi 3520	1 |- ( Clsd ` { (/) , A } ) = { (/) , A }

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   \ cdif 3473   C_ wss 3476   (/)c0 3785   {cpr 4029   _I cid 4790   ` cfv 5586   Topctop 19158   Clsdccld 19280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19163  df-topon 19166  df-cld 19283

This theorem is referenced by: (None)