Theorem fbasfip 21672

Description: A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fbasfip	|- ( F e. ( fBas ` X ) -> -. (/) e. ( fi ` F ) )

Proof of Theorem fbasfip

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . 6 |- ( y e. ( ~P F i^i Fin ) <-> ( y e. ~P F /\ y e. Fin ) )
2		elpwi 4168	. . . . . . 7 |- ( y e. ~P F -> y C_ F )
3	2	anim1i 592	. . . . . 6 |- ( ( y e. ~P F /\ y e. Fin ) -> ( y C_ F /\ y e. Fin ) )
4	1, 3	sylbi 207	. . . . 5 |- ( y e. ( ~P F i^i Fin ) -> ( y C_ F /\ y e. Fin ) )
5		fbssint 21642	. . . . . 6 |- ( ( F e. ( fBas ` X ) /\ y C_ F /\ y e. Fin ) -> E. z e. F z C_ |^| y )
6	5	3expb 1266	. . . . 5 |- ( ( F e. ( fBas ` X ) /\ ( y C_ F /\ y e. Fin ) ) -> E. z e. F z C_ |^| y )
7	4, 6	sylan2 491	. . . 4 |- ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) -> E. z e. F z C_ |^| y )
8		0nelfb 21635	. . . . . . . . 9 |- ( F e. ( fBas ` X ) -> -. (/) e. F )
9	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ z e. F ) -> -. (/) e. F )
10		eleq1 2689	. . . . . . . . . 10 |- ( z = (/) -> ( z e. F <-> (/) e. F ) )
11	10	biimpcd 239	. . . . . . . . 9 |- ( z e. F -> ( z = (/) -> (/) e. F ) )
12	11	adantl 482	. . . . . . . 8 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ z e. F ) -> ( z = (/) -> (/) e. F ) )
13	9, 12	mtod 189	. . . . . . 7 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ z e. F ) -> -. z = (/) )
14		ss0 3974	. . . . . . 7 |- ( z C_ (/) -> z = (/) )
15	13, 14	nsyl 135	. . . . . 6 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ z e. F ) -> -. z C_ (/) )
16	15	adantrr 753	. . . . 5 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ ( z e. F /\ z C_ |^| y ) ) -> -. z C_ (/) )
17		sseq2 3627	. . . . . . 7 |- ( (/) = |^| y -> ( z C_ (/) <-> z C_ |^| y ) )
18	17	biimprcd 240	. . . . . 6 |- ( z C_ |^| y -> ( (/) = |^| y -> z C_ (/) ) )
19	18	ad2antll 765	. . . . 5 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ ( z e. F /\ z C_ |^| y ) ) -> ( (/) = |^| y -> z C_ (/) ) )
20	16, 19	mtod 189	. . . 4 |- ( ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) /\ ( z e. F /\ z C_ |^| y ) ) -> -. (/) = |^| y )
21	7, 20	rexlimddv 3035	. . 3 |- ( ( F e. ( fBas ` X ) /\ y e. ( ~P F i^i Fin ) ) -> -. (/) = |^| y )
22	21	nrexdv 3001	. 2 |- ( F e. ( fBas ` X ) -> -. E. y e. ( ~P F i^i Fin ) (/) = |^| y )
23		0ex 4790	. . 3 |- (/) e. _V
24		elfi 8319	. . 3 |- ( ( (/) e. _V /\ F e. ( fBas ` X ) ) -> ( (/) e. ( fi ` F ) <-> E. y e. ( ~P F i^i Fin ) (/) = |^| y ) )
25	23, 24	mpan 706	. 2 |- ( F e. ( fBas ` X ) -> ( (/) e. ( fi ` F ) <-> E. y e. ( ~P F i^i Fin ) (/) = |^| y ) )
26	22, 25	mtbird 315	1 |- ( F e. ( fBas ` X ) -> -. (/) e. ( fi ` F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888   Fincfn 7955   ficfi 8316   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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743

This theorem is referenced by:  fbunfip  21673