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Theorem snfbas 17561
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 4160 . . . . 5  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
213adant2 974 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  _V )
3 simp2 956 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  =/=  (/) )
4 snfil 17559 . . . 4  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
52, 3, 4syl2anc 642 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( Fil `  A ) )
6 filfbas 17543 . . 3  |-  ( { A }  e.  ( Fil `  A )  ->  { A }  e.  ( fBas `  A
) )
75, 6syl 15 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  A ) )
8 simp1 955 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  C_  B )
9 elpw2g 4174 . . . . 5  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
1093ad2ant3 978 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
118, 10mpbird 223 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  ~P B )
1211snssd 3760 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  C_  ~P B )
13 simp3 957 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  B  e.  V )
14 fbasweak 17560 . 2  |-  ( ( { A }  e.  ( fBas `  A )  /\  { A }  C_  ~P B  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
157, 12, 13, 14syl3anc 1182 1  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5255   fBascfbas 17518   Filcfil 17540
This theorem is referenced by:  isufil2  17603  ufileu  17614  filufint  17615  uffix  17616  flimclslem  17679  cnfilca  25556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520  df-fil 17541
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