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Theorem snfbas 20102
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 4593 . . . . 5  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
213adant2 1015 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  _V )
3 simp2 997 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  =/=  (/) )
4 snfil 20100 . . . 4  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
52, 3, 4syl2anc 661 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( Fil `  A ) )
6 filfbas 20084 . . 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 996 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  C_  B )
9 elpw2g 4610 . . . . 5  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
1093ad2ant3 1019 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
118, 10mpbird 232 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  ~P B )
1211snssd 4172 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  C_  ~P B )
13 simp3 998 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  B  e.  V )
14 fbasweak 20101 . 2  |-  ( ( { A }  e.  ( fBas `  A )  /\  { A }  C_  ~P B  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
157, 12, 13, 14syl3anc 1228 1  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   ` cfv 5586   fBascfbas 18177   Filcfil 20081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-fbas 18187  df-fil 20082
This theorem is referenced by:  isufil2  20144  ufileu  20155  filufint  20156  uffix  20157  flimclslem  20220
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