Theorem snfbas 21670

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

Step	Hyp	Ref	Expression
1		ssexg 4804	. . . . 5 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2	1	3adant2 1080	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> A e. _V )
3		simp2 1062	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> A =/= (/) )
4		snfil 21668	. . . 4 |- ( ( A e. _V /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )
5	2, 3, 4	syl2anc 693	. . 3 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> { A } e. ( Fil ` A ) )
6		filfbas 21652	. . 3 |- ( { A } e. ( Fil ` A ) -> { A } e. ( fBas ` A ) )
7	5, 6	syl 17	. 2 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> { A } e. ( fBas ` A ) )
8		simp1 1061	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> A C_ B )
9		elpw2g 4827	. . . . 5 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )
10	9	3ad2ant3 1084	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> ( A e. ~P B <-> A C_ B ) )
11	8, 10	mpbird 247	. . 3 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> A e. ~P B )
12	11	snssd 4340	. 2 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> { A } C_ ~P B )
13		simp3 1063	. 2 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> B e. V )
14		fbasweak 21669	. 2 |- ( ( { A } e. ( fBas ` A ) /\ { A } C_ ~P B /\ B e. V ) -> { A } e. ( fBas ` B ) )
15	7, 12, 13, 14	syl3anc 1326	1 |- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> { A } e. ( fBas ` B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   ` cfv 5888   fBascfbas 19734   Filcfil 21649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-fbas 19743  df-fil 21650

This theorem is referenced by:  isufil2  21712  ufileu  21723  filufint  21724  uffix  21725  flimclslem  21788