Theorem filss 20868

Description: A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Assertion

Ref	Expression
filss	|- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. F )

Proof of Theorem filss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfil 20862	. . . 4 |- ( F e. ( Fil ` X ) <-> ( F e. ( fBas ` X ) /\ A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )
2	1	simprbi 466	. . 3 |- ( F e. ( Fil ` X ) -> A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) )
3	2	adantr 467	. 2 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) )
4		elfvdm 5891	. . 3 |- ( F e. ( Fil ` X ) -> X e. dom Fil )
5		simp2 1009	. . 3 |- ( ( A e. F /\ B C_ X /\ A C_ B ) -> B C_ X )
6		elpw2g 4566	. . . 4 |- ( X e. dom Fil -> ( B e. ~P X <-> B C_ X ) )
7	6	biimpar 488	. . 3 |- ( ( X e. dom Fil /\ B C_ X ) -> B e. ~P X )
8	4, 5, 7	syl2an 480	. 2 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. ~P X )
9		simpr1 1014	. . 3 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A e. F )
10		simpr3 1016	. . . 4 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A C_ B )
11		elpwg 3959	. . . . 5 |- ( A e. F -> ( A e. ~P B <-> A C_ B ) )
12	9, 11	syl 17	. . . 4 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> ( A e. ~P B <-> A C_ B ) )
13	10, 12	mpbird 236	. . 3 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A e. ~P B )
14		inelcm 3819	. . 3 |- ( ( A e. F /\ A e. ~P B ) -> ( F i^i ~P B ) =/= (/) )
15	9, 13, 14	syl2anc 667	. 2 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> ( F i^i ~P B ) =/= (/) )
16		pweq 3954	. . . . . 6 |- ( x = B -> ~P x = ~P B )
17	16	ineq2d 3634	. . . . 5 |- ( x = B -> ( F i^i ~P x ) = ( F i^i ~P B ) )
18	17	neeq1d 2683	. . . 4 |- ( x = B -> ( ( F i^i ~P x ) =/= (/) <-> ( F i^i ~P B ) =/= (/) ) )
19		eleq1 2517	. . . 4 |- ( x = B -> ( x e. F <-> B e. F ) )
20	18, 19	imbi12d 322	. . 3 |- ( x = B -> ( ( ( F i^i ~P x ) =/= (/) -> x e. F ) <-> ( ( F i^i ~P B ) =/= (/) -> B e. F ) ) )
21	20	rspccv 3147	. 2 |- ( A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) -> ( B e. ~P X -> ( ( F i^i ~P B ) =/= (/) -> B e. F ) ) )
22	3, 8, 15, 21	syl3c 63	1 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. F )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   i^i cin 3403   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   dom cdm 4834   ` cfv 5582   fBascfbas 18958   Filcfil 20860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-fil 20861

This theorem is referenced by:  filin  20869  filtop  20870  isfil2  20871  infil  20878  fgfil  20890  fgabs  20894  filcon  20898  filuni  20900  trfil2  20902  trfg  20906  isufil2  20923  ufprim  20924  ufileu  20934  filufint  20935  elfm3  20965  rnelfm  20968  fmfnfmlem2  20970  fmfnfmlem4  20972  flimopn  20990  flimrest  20998  flimfnfcls  21043  fclscmpi  21044  alexsublem  21059  metust  21573  cfil3i  22239  cfilfcls  22244  iscmet3lem2  22262  equivcfil  22269  relcmpcmet  22286  minveclem4  22374  minveclem4OLD  22386  fgmin  31026