Theorem fipfil2 10272

Description: A filter has the finite intersection property. Bourbaki TG I.36 note of def. 1. (Contributed by FL, 2-Sep-2007.)

Assertion

Ref	Expression
fipfil2	|- (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> |^|A =/= (/)))

Proof of Theorem fipfil2

Step	Hyp	Ref	Expression
1		ssexg 3457	. . . . . . . . . 10 |- ((A C_ F /\ F e. Fil) -> A e. _V)
2		sseq1 2637	. . . . . . . . . . . . . . 15 |- (y = A -> (y C_ F <-> A C_ F))
3		neeq1 2024	. . . . . . . . . . . . . . 15 |- (y = A -> (y =/= (/) <-> A =/= (/)))
4		eleq1 1957	. . . . . . . . . . . . . . 15 |- (y = A -> (y e. Fin <-> A e. Fin))
5	2, 3, 4	3anbi123d 1168	. . . . . . . . . . . . . 14 |- (y = A -> ((y C_ F /\ y =/= (/) /\ y e. Fin) <-> (A C_ F /\ A =/= (/) /\ A e. Fin)))
6		inteq 3217	. . . . . . . . . . . . . . 15 |- (y = A -> |^|y = |^|A)
7		eqeq1 1890	. . . . . . . . . . . . . . . 16 |- (|^|y = |^|A -> (|^|y = (/) <-> |^|A = (/)))
8	7	notbid 673	. . . . . . . . . . . . . . 15 |- (|^|y = |^|A -> (-. |^|y = (/) <-> -. |^|A = (/)))
9	6, 8	syl 12	. . . . . . . . . . . . . 14 |- (y = A -> (-. |^|y = (/) <-> -. |^|A = (/)))
10	5, 9	imbi12d 688	. . . . . . . . . . . . 13 |- (y = A -> (((y C_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)) <-> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
11	10	cla4gv 2364	. . . . . . . . . . . 12 |- (A e. _V -> (A.y((y C_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)) -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
12		eqid 1884	. . . . . . . . . . . . . . . . . . . . . 22 |- U.F = U.F
13	12	isfil 10266	. . . . . . . . . . . . . . . . . . . . 21 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x C_ U.F /\ y C_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F)))
14	13	biimpa 460	. . . . . . . . . . . . . . . . . . . 20 |- ((F e. Fil /\ F e. Fil) -> ((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x C_ U.F /\ y C_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F))
15	14	simp3d 890	. . . . . . . . . . . . . . . . . . 19 |- ((F e. Fil /\ F e. Fil) -> A.y e. F A.x e. F (y i^i x) e. F)
16	15	anidms 480	. . . . . . . . . . . . . . . . . 18 |- (F e. Fil -> A.y e. F A.x e. F (y i^i x) e. F)
17		fiint 5650	. . . . . . . . . . . . . . . . . 18 |- (A.y e. F A.x e. F (y i^i x) e. F <-> A.y((y C_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
18	16, 17	sylib 215	. . . . . . . . . . . . . . . . 17 |- (F e. Fil -> A.y((y C_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
19	18	19.21bi 1408	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> ((y C_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
20	19	imp 377	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y C_ F /\ y =/= (/) /\ y e. Fin)) -> |^|y e. F)
21		filesn 10268	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> -. (/) e. F)
22	21	adantr 425	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y C_ F /\ y =/= (/) /\ y e. Fin)) -> -. (/) e. F)
23		nelneq 1985	. . . . . . . . . . . . . . 15 |- ((|^|y e. F /\ -. (/) e. F) -> -. |^|y = (/))
24	20, 22, 23	syl11anc 524	. . . . . . . . . . . . . 14 |- ((F e. Fil /\ (y C_ F /\ y =/= (/) /\ y e. Fin)) -> -. |^|y = (/))
25	24	ex 402	. . . . . . . . . . . . 13 |- (F e. Fil -> ((y C_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)))
26	25	19.21aiv 1664	. . . . . . . . . . . 12 |- (F e. Fil -> A.y((y C_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)))
27	11, 26	syl5 20	. . . . . . . . . . 11 |- (A e. _V -> (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
28	27	com23 36	. . . . . . . . . 10 |- (A e. _V -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
29	1, 28	syl 12	. . . . . . . . 9 |- ((A C_ F /\ F e. Fil) -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
30	29	ex 402	. . . . . . . 8 |- (A C_ F -> (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/)))))
31	30	com14 42	. . . . . . 7 |- (F e. Fil -> (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (A C_ F -> -. |^|A = (/)))))
32	31	pm2.43i 78	. . . . . 6 |- (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (A C_ F -> -. |^|A = (/))))
33	32	com13 37	. . . . 5 |- (A C_ F -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
34	33	3ad2ant1 897	. . . 4 |- ((A C_ F /\ A =/= (/) /\ A e. Fin) -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
35	34	pm2.43i 78	. . 3 |- ((A C_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/)))
36	35	com12 14	. 2 |- (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/)))
37		df-ne 2019	. 2 |- (|^|A =/= (/) <-> -. |^|A = (/))
38	36, 37	syl6ibr 230	1 |- (F e. Fil -> ((A C_ F /\ A =/= (/) /\ A e. Fin) -> |^|A =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214  Fincfn 5426  Filcfil 10264

This theorem is referenced by:  efilcp 14922  efilcp2 14926  cnfilca 14927

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430  df-fil 10265

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