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Theorem snfbas 17577
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 4176 . . . . 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 17575 . . . 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 17559 . . 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 4190 . . . . 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 3776 . 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 17576 . 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 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   ` cfv 5271   fBascfbas 17534   Filcfil 17556
This theorem is referenced by:  isufil2  17619  ufileu  17630  filufint  17631  uffix  17632  flimclslem  17695  cnfilca  25659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-fbas 17536  df-fil 17557
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