Theorem mreintcl 16255

Description: A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mreintcl	|- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> |^| S e. C )

Proof of Theorem mreintcl

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw2g 4827	. . . 4 |- ( C e. (Moore ` X ) -> ( S e. ~P C <-> S C_ C ) )
2	1	biimpar 502	. . 3 |- ( ( C e. (Moore ` X ) /\ S C_ C ) -> S e. ~P C )
3	2	3adant3 1081	. 2 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> S e. ~P C )
4		ismre 16250	. . . 4 |- ( C e. (Moore ` X ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) )
5	4	simp3bi 1078	. . 3 |- ( C e. (Moore ` X ) -> A. s e. ~P C ( s =/= (/) -> |^| s e. C ) )
6	5	3ad2ant1 1082	. 2 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> A. s e. ~P C ( s =/= (/) -> |^| s e. C ) )
7		simp3 1063	. 2 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> S =/= (/) )
8		neeq1 2856	. . . . 5 |- ( s = S -> ( s =/= (/) <-> S =/= (/) ) )
9		inteq 4478	. . . . . 6 |- ( s = S -> |^| s = |^| S )
10	9	eleq1d 2686	. . . . 5 |- ( s = S -> ( |^| s e. C <-> |^| S e. C ) )
11	8, 10	imbi12d 334	. . . 4 |- ( s = S -> ( ( s =/= (/) -> |^| s e. C ) <-> ( S =/= (/) -> |^| S e. C ) ) )
12	11	rspcva 3307	. . 3 |- ( ( S e. ~P C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) -> ( S =/= (/) -> |^| S e. C ) )
13	12	3impia 1261	. 2 |- ( ( S e. ~P C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) /\ S =/= (/) ) -> |^| S e. C )
14	3, 6, 7, 13	syl3anc 1326	1 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> |^| S e. C )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888  Moorecmre 16242

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

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-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  mreiincl  16256  mrerintcl  16257  mreincl  16259  mremre  16264  submre  16265  mrcflem  16266  mrelatglb  17184  mreclatBAD  17187