Theorem indiscld 18654

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 18565	. . . . 5 |- { (/) , A } e. Top
2		indisuni 18566	. . . . . 6 |- ( _I ` A ) = U. { (/) , A }
3	2	iscld 18590	. . . . 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 454	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> x C_ ( _I ` A ) )
6		dfss4 3581	. . . . . 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 458	. . . . . . 7 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , A } )
9		indislem 18563	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
10	8, 9	syl6eleqr 2532	. . . . . 6 |- ( ( x C_ ( _I ` A ) /\ ( ( _I ` A ) \ x ) e. { (/) , A } ) -> ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } )
11		elpri 3894	. . . . . 6 |- ( ( ( _I ` A ) \ x ) e. { (/) , ( _I ` A ) } -> ( ( ( _I ` A ) \ x ) = (/) \/ ( ( _I ` A ) \ x ) = ( _I ` A ) ) )
12		difeq2 3465	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ (/) ) )
13		dif0 3746	. . . . . . . . 9 |- ( ( _I ` A ) \ (/) ) = ( _I ` A )
14	12, 13	syl6eq 2489	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( _I ` A ) )
15		fvex 5698	. . . . . . . . . 10 |- ( _I ` A ) e. _V
16	15	prid2 3981	. . . . . . . . 9 |- ( _I ` A ) e. { (/) , ( _I ` A ) }
17	16, 9	eleqtri 2513	. . . . . . . 8 |- ( _I ` A ) e. { (/) , A }
18	14, 17	syl6eqel 2529	. . . . . . 7 |- ( ( ( _I ` A ) \ x ) = (/) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) e. { (/) , A } )
19		difeq2 3465	. . . . . . . . 9 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = ( ( _I ` A ) \ ( _I ` A ) ) )
20		difid 3744	. . . . . . . . 9 |- ( ( _I ` A ) \ ( _I ` A ) ) = (/)
21	19, 20	syl6eq 2489	. . . . . . . 8 |- ( ( ( _I ` A ) \ x ) = ( _I ` A ) -> ( ( _I ` A ) \ ( ( _I ` A ) \ x ) ) = (/) )
22		0ex 4419	. . . . . . . . 9 |- (/) e. _V
23	22	prid1 3980	. . . . . . . 8 |- (/) e. { (/) , A }
24	21, 23	syl6eqel 2529	. . . . . . 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 2516	. . . 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 3357	. 2 |- ( Clsd ` { (/) , A } ) C_ { (/) , A }
30		0cld 18601	. . . . 5 |- ( { (/) , A } e. Top -> (/) e. ( Clsd ` { (/) , A } ) )
31	1, 30	ax-mp 5	. . . 4 |- (/) e. ( Clsd ` { (/) , A } )
32	2	topcld 18598	. . . . 5 |- ( { (/) , A } e. Top -> ( _I ` A ) e. ( Clsd ` { (/) , A } ) )
33	1, 32	ax-mp 5	. . . 4 |- ( _I ` A ) e. ( Clsd ` { (/) , A } )
34		prssi 4026	. . . 4 |- ( ( (/) e. ( Clsd ` { (/) , A } ) /\ ( _I ` A ) e. ( Clsd ` { (/) , A } ) ) -> { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } ) )
35	31, 33, 34	mp2an 667	. . 3 |- { (/) , ( _I ` A ) } C_ ( Clsd ` { (/) , A } )
36	9, 35	eqsstr3i 3384	. 2 |- { (/) , A } C_ ( Clsd ` { (/) , A } )
37	29, 36	eqssi 3369	1 |- ( Clsd ` { (/) , A } ) = { (/) , A }

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1364   e. wcel 1761   \ cdif 3322   C_ wss 3325   (/)c0 3634   {cpr 3876   _I cid 4627   ` cfv 5415   Topctop 18457   Clsdccld 18579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-top 18462  df-topon 18465  df-cld 18582

This theorem is referenced by: (None)