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Theorem snfbas 17888
Description: Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
snfbas  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )

Proof of Theorem snfbas
StepHypRef Expression
1 ssexg 4341 . . . . 5  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
213adant2 976 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  _V )
3 simp2 958 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  =/=  (/) )
4 snfil 17886 . . . 4  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
52, 3, 4syl2anc 643 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( Fil `  A ) )
6 filfbas 17870 . . 3  |-  ( { A }  e.  ( Fil `  A )  ->  { A }  e.  ( fBas `  A
) )
75, 6syl 16 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  A ) )
8 simp1 957 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  C_  B )
9 elpw2g 4355 . . . . 5  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
1093ad2ant3 980 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
118, 10mpbird 224 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  ~P B )
1211snssd 3935 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  C_  ~P B )
13 simp3 959 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  B  e.  V )
14 fbasweak 17887 . 2  |-  ( ( { A }  e.  ( fBas `  A )  /\  { A }  C_  ~P B  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
157, 12, 13, 14syl3anc 1184 1  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   ` cfv 5446   fBascfbas 16679   Filcfil 17867
This theorem is referenced by:  isufil2  17930  ufileu  17941  filufint  17942  uffix  17943  flimclslem  18006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-fbas 16689  df-fil 17868
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