Theorem filss 21657

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 21651	. . . 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 480	. . 3 |- ( F e. ( Fil ` X ) -> A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) )
3	2	adantr 481	. 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 6220	. . 3 |- ( F e. ( Fil ` X ) -> X e. dom Fil )
5		simp2 1062	. . 3 |- ( ( A e. F /\ B C_ X /\ A C_ B ) -> B C_ X )
6		elpw2g 4827	. . . 4 |- ( X e. dom Fil -> ( B e. ~P X <-> B C_ X ) )
7	6	biimpar 502	. . 3 |- ( ( X e. dom Fil /\ B C_ X ) -> B e. ~P X )
8	4, 5, 7	syl2an 494	. 2 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. ~P X )
9		simpr1 1067	. . 3 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A e. F )
10		simpr3 1069	. . . 4 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A C_ B )
11		elpwg 4166	. . . . 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 247	. . 3 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> A e. ~P B )
14		inelcm 4032	. . 3 |- ( ( A e. F /\ A e. ~P B ) -> ( F i^i ~P B ) =/= (/) )
15	9, 13, 14	syl2anc 693	. 2 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> ( F i^i ~P B ) =/= (/) )
16		pweq 4161	. . . . . 6 |- ( x = B -> ~P x = ~P B )
17	16	ineq2d 3814	. . . . 5 |- ( x = B -> ( F i^i ~P x ) = ( F i^i ~P B ) )
18	17	neeq1d 2853	. . . 4 |- ( x = B -> ( ( F i^i ~P x ) =/= (/) <-> ( F i^i ~P B ) =/= (/) ) )
19		eleq1 2689	. . . 4 |- ( x = B -> ( x e. F <-> B e. F ) )
20	18, 19	imbi12d 334	. . 3 |- ( x = B -> ( ( ( F i^i ~P x ) =/= (/) -> x e. F ) <-> ( ( F i^i ~P B ) =/= (/) -> B e. F ) ) )
21	20	rspccv 3306	. 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 66	1 |- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   fBascfbas 19734   Filcfil 21649

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-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-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-iota 5851  df-fun 5890  df-fv 5896  df-fil 21650

This theorem is referenced by:  filin  21658  filtop  21659  isfil2  21660  infil  21667  fgfil  21679  fgabs  21683  filconn  21687  filuni  21689  trfil2  21691  trfg  21695  isufil2  21712  ufprim  21713  ufileu  21723  filufint  21724  elfm3  21754  rnelfm  21757  fmfnfmlem2  21759  fmfnfmlem4  21761  flimopn  21779  flimrest  21787  flimfnfcls  21832  fclscmpi  21833  alexsublem  21848  metust  22363  cfil3i  23067  cfilfcls  23072  iscmet3lem2  23090  equivcfil  23097  relcmpcmet  23115  minveclem4  23203  fgmin  32365