Theorem cmpfii 21212

Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
cmpfii	|- ( ( J e. Comp /\ X C_ ( Clsd ` J ) /\ -. (/) e. ( fi ` X ) ) -> |^| X =/= (/) )

Proof of Theorem cmpfii

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- ( Clsd ` J ) e. _V
2	1	elpw2 4828	. . . 4 |- ( X e. ~P ( Clsd ` J ) <-> X C_ ( Clsd ` J ) )
3	2	biimpri 218	. . 3 |- ( X C_ ( Clsd ` J ) -> X e. ~P ( Clsd ` J ) )
4		cmptop 21198	. . . . 5 |- ( J e. Comp -> J e. Top )
5		cmpfi 21211	. . . . 5 |- ( J e. Top -> ( J e. Comp <-> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) )
6	4, 5	syl 17	. . . 4 |- ( J e. Comp -> ( J e. Comp <-> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) )
7	6	ibi 256	. . 3 |- ( J e. Comp -> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) )
8		fveq2 6191	. . . . . . 7 |- ( x = X -> ( fi ` x ) = ( fi ` X ) )
9	8	eleq2d 2687	. . . . . 6 |- ( x = X -> ( (/) e. ( fi ` x ) <-> (/) e. ( fi ` X ) ) )
10	9	notbid 308	. . . . 5 |- ( x = X -> ( -. (/) e. ( fi ` x ) <-> -. (/) e. ( fi ` X ) ) )
11		inteq 4478	. . . . . 6 |- ( x = X -> |^| x = |^| X )
12	11	neeq1d 2853	. . . . 5 |- ( x = X -> ( |^| x =/= (/) <-> |^| X =/= (/) ) )
13	10, 12	imbi12d 334	. . . 4 |- ( x = X -> ( ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) <-> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) ) )
14	13	rspcva 3307	. . 3 |- ( ( X e. ~P ( Clsd ` J ) /\ A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) -> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) )
15	3, 7, 14	syl2anr 495	. 2 |- ( ( J e. Comp /\ X C_ ( Clsd ` J ) ) -> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) )
16	15	3impia 1261	1 |- ( ( J e. Comp /\ X C_ ( Clsd ` J ) /\ -. (/) e. ( fi ` X ) ) -> |^| X =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888   ficfi 8316   Topctop 20698   Clsdccld 20820   Compccmp 21189

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  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-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-top 20699  df-cld 20823  df-cmp 21190

This theorem is referenced by:  fclscmpi  21833  cmpfiiin  37260