Theorem t1connperf 21239

Description: A connected T₁ space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)

Hypothesis

Ref	Expression
t1connperf.1	|- X = U. J

Assertion

Ref	Expression
t1connperf	|- ( ( J e. Fre /\ J e. Conn /\ -. X ~~ 1o ) -> J e. Perf )

Proof of Theorem t1connperf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		t1connperf.1	. . . . . . . 8 |- X = U. J
2		simplr 792	. . . . . . . 8 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> J e. Conn )
3		simprr 796	. . . . . . . 8 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> { x } e. J )
4		vex 3203	. . . . . . . . . 10 |- x e. _V
5	4	snnz 4309	. . . . . . . . 9 |- { x } =/= (/)
6	5	a1i 11	. . . . . . . 8 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> { x } =/= (/) )
7	1	t1sncld 21130	. . . . . . . . 9 |- ( ( J e. Fre /\ x e. X ) -> { x } e. ( Clsd ` J ) )
8	7	ad2ant2r 783	. . . . . . . 8 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> { x } e. ( Clsd ` J ) )
9	1, 2, 3, 6, 8	connclo 21218	. . . . . . 7 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> { x } = X )
10	4	ensn1 8020	. . . . . . 7 |- { x } ~~ 1o
11	9, 10	syl6eqbrr 4693	. . . . . 6 |- ( ( ( J e. Fre /\ J e. Conn ) /\ ( x e. X /\ { x } e. J ) ) -> X ~~ 1o )
12	11	rexlimdvaa 3032	. . . . 5 |- ( ( J e. Fre /\ J e. Conn ) -> ( E. x e. X { x } e. J -> X ~~ 1o ) )
13	12	con3d 148	. . . 4 |- ( ( J e. Fre /\ J e. Conn ) -> ( -. X ~~ 1o -> -. E. x e. X { x } e. J ) )
14		ralnex 2992	. . . 4 |- ( A. x e. X -. { x } e. J <-> -. E. x e. X { x } e. J )
15	13, 14	syl6ibr 242	. . 3 |- ( ( J e. Fre /\ J e. Conn ) -> ( -. X ~~ 1o -> A. x e. X -. { x } e. J ) )
16		t1top 21134	. . . . 5 |- ( J e. Fre -> J e. Top )
17	16	adantr 481	. . . 4 |- ( ( J e. Fre /\ J e. Conn ) -> J e. Top )
18	1	isperf3 20957	. . . . 5 |- ( J e. Perf <-> ( J e. Top /\ A. x e. X -. { x } e. J ) )
19	18	baib 944	. . . 4 |- ( J e. Top -> ( J e. Perf <-> A. x e. X -. { x } e. J ) )
20	17, 19	syl 17	. . 3 |- ( ( J e. Fre /\ J e. Conn ) -> ( J e. Perf <-> A. x e. X -. { x } e. J ) )
21	15, 20	sylibrd 249	. 2 |- ( ( J e. Fre /\ J e. Conn ) -> ( -. X ~~ 1o -> J e. Perf ) )
22	21	3impia 1261	1 |- ( ( J e. Fre /\ J e. Conn /\ -. X ~~ 1o ) -> J e. Perf )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   ` cfv 5888   1oc1o 7553   ~~ cen 7952   Topctop 20698   Clsdccld 20820  Perfcperf 20939   Frect1 21111  Conncconn 21214

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  df-iun 4522  df-iin 4523  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-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-en 7956  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825  df-lp 20940  df-perf 20941  df-t1 21118  df-conn 21215

This theorem is referenced by: (None)