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Theorem snfbas 19557
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 4538 . . . . 5  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
213adant2 1007 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  e.  _V )
3 simp2 989 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  =/=  (/) )
4 snfil 19555 . . . 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 19539 . . 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 988 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  A  C_  B )
9 elpw2g 4555 . . . . 5  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
1093ad2ant3 1011 . . . 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 4118 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  C_  ~P B )
13 simp3 990 . 2  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  B  e.  V )
14 fbasweak 19556 . 2  |-  ( ( { A }  e.  ( fBas `  A )  /\  { A }  C_  ~P B  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
157, 12, 13, 14syl3anc 1219 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 965    e. wcel 1758    =/= wne 2644   _Vcvv 3070    C_ wss 3428   (/)c0 3737   ~Pcpw 3960   {csn 3977   ` cfv 5518   fBascfbas 17915   Filcfil 19536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-fbas 17925  df-fil 19537
This theorem is referenced by:  isufil2  19599  ufileu  19610  filufint  19611  uffix  19612  flimclslem  19675
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