Theorem ufilb 21710

Description: The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)

Assertion

Ref	Expression
ufilb	|- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( -. S e. F <-> ( X \ S ) e. F ) )

Proof of Theorem ufilb

Step	Hyp	Ref	Expression
1		ufilss 21709	. . 3 |- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( S e. F \/ ( X \ S ) e. F ) )
2	1	ord 392	. 2 |- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( -. S e. F -> ( X \ S ) e. F ) )
3		ufilfil 21708	. . . 4 |- ( F e. ( UFil ` X ) -> F e. ( Fil ` X ) )
4		filfbas 21652	. . . 4 |- ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
5		fbncp 21643	. . . . . 6 |- ( ( F e. ( fBas ` X ) /\ S e. F ) -> -. ( X \ S ) e. F )
6	5	ex 450	. . . . 5 |- ( F e. ( fBas ` X ) -> ( S e. F -> -. ( X \ S ) e. F ) )
7	6	con2d 129	. . . 4 |- ( F e. ( fBas ` X ) -> ( ( X \ S ) e. F -> -. S e. F ) )
8	3, 4, 7	3syl 18	. . 3 |- ( F e. ( UFil ` X ) -> ( ( X \ S ) e. F -> -. S e. F ) )
9	8	adantr 481	. 2 |- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( ( X \ S ) e. F -> -. S e. F ) )
10	2, 9	impbid 202	1 |- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( -. S e. F <-> ( X \ S ) e. F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   \ cdif 3571   C_ wss 3574   ` cfv 5888   fBascfbas 19734   Filcfil 21649   UFilcufil 21703

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  df-ufil 21705

This theorem is referenced by:  ufilmax  21711  ufprim  21713  trufil  21714  ufileu  21723  cfinufil  21732  alexsublem  21848