Theorem fbncp 21643

Description: A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Assertion

Ref	Expression
fbncp	|- ( ( F e. ( fBas ` X ) /\ A e. F ) -> -. ( B \ A ) e. F )

Proof of Theorem fbncp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelfb 21635	. . 3 |- ( F e. ( fBas ` X ) -> -. (/) e. F )
2	1	adantr 481	. 2 |- ( ( F e. ( fBas ` X ) /\ A e. F ) -> -. (/) e. F )
3		fbasssin 21640	. . . 4 |- ( ( F e. ( fBas ` X ) /\ A e. F /\ ( B \ A ) e. F ) -> E. x e. F x C_ ( A i^i ( B \ A ) ) )
4		disjdif 4040	. . . . . . . 8 |- ( A i^i ( B \ A ) ) = (/)
5	4	sseq2i 3630	. . . . . . 7 |- ( x C_ ( A i^i ( B \ A ) ) <-> x C_ (/) )
6		ss0 3974	. . . . . . 7 |- ( x C_ (/) -> x = (/) )
7	5, 6	sylbi 207	. . . . . 6 |- ( x C_ ( A i^i ( B \ A ) ) -> x = (/) )
8		eleq1 2689	. . . . . . 7 |- ( x = (/) -> ( x e. F <-> (/) e. F ) )
9	8	biimpac 503	. . . . . 6 |- ( ( x e. F /\ x = (/) ) -> (/) e. F )
10	7, 9	sylan2 491	. . . . 5 |- ( ( x e. F /\ x C_ ( A i^i ( B \ A ) ) ) -> (/) e. F )
11	10	rexlimiva 3028	. . . 4 |- ( E. x e. F x C_ ( A i^i ( B \ A ) ) -> (/) e. F )
12	3, 11	syl 17	. . 3 |- ( ( F e. ( fBas ` X ) /\ A e. F /\ ( B \ A ) e. F ) -> (/) e. F )
13	12	3expia 1267	. 2 |- ( ( F e. ( fBas ` X ) /\ A e. F ) -> ( ( B \ A ) e. F -> (/) e. F ) )
14	2, 13	mtod 189	1 |- ( ( F e. ( fBas ` X ) /\ A e. F ) -> -. ( B \ A ) e. F )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ` cfv 5888   fBascfbas 19734

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

This theorem is referenced by:  filconn  21687  fgtr  21694  ufilb  21710  ufilmax  21711  ufilen  21734  flimrest  21787  fclsrest  21828  cfilres  23094  relcmpcmet  23115