Theorem fipfil2 10550

Description: A filter has the finite intersection property. Bourbaki TG I.36 note of def. 1.

Assertion

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

Proof of Theorem fipfil2

Step	Hyp	Ref	Expression
1		ssexg 2726	. . . . . . . . . 10 |- ((A (_ F /\ F e. Fil) -> A e. V)
2		sseq1 2085	. . . . . . . . . . . . . . 15 |- (y = A -> (y (_ F <-> A (_ F))
3		neeq1 1593	. . . . . . . . . . . . . . 15 |- (y = A -> (y =/= (/) <-> A =/= (/)))
4		eleq1 1537	. . . . . . . . . . . . . . 15 |- (y = A -> (y e. Fin <-> A e. Fin))
5	2, 3, 4	3anbi123d 895	. . . . . . . . . . . . . 14 |- (y = A -> ((y (_ F /\ y =/= (/) /\ y e. Fin) <-> (A (_ F /\ A =/= (/) /\ A e. Fin)))
6		inteq 2540	. . . . . . . . . . . . . . 15 |- (y = A -> |^|y = |^|A)
7		eqeq1 1484	. . . . . . . . . . . . . . . 16 |- (|^|y = |^|A -> (|^|y = (/) <-> |^|A = (/)))
8	7	negbid 613	. . . . . . . . . . . . . . 15 |- (|^|y = |^|A -> (-. |^|y = (/) <-> -. |^|A = (/)))
9	6, 8	syl 10	. . . . . . . . . . . . . 14 |- (y = A -> (-. |^|y = (/) <-> -. |^|A = (/)))
10	5, 9	imbi12d 628	. . . . . . . . . . . . 13 |- (y = A -> (((y (_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)) <-> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
11	10	cla4gv 1865	. . . . . . . . . . . 12 |- (A e. V -> (A.y((y (_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)) -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
12		nelneq 1564	. . . . . . . . . . . . . . 15 |- ((|^|y e. F /\ -. (/) e. F) -> -. |^|y = (/))
13		eqid 1478	. . . . . . . . . . . . . . . . . . . . . 22 |- U.F = U.F
14	13	isfil 10544	. . . . . . . . . . . . . . . . . . . . 21 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F)))
15	14	biimpa 418	. . . . . . . . . . . . . . . . . . . 20 |- ((F e. Fil /\ F e. Fil) -> ((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F))
16	15	3simp3d 798	. . . . . . . . . . . . . . . . . . 19 |- ((F e. Fil /\ F e. Fil) -> A.y e. F A.x e. F (y i^i x) e. F)
17	16	anidms 436	. . . . . . . . . . . . . . . . . 18 |- (F e. Fil -> A.y e. F A.x e. F (y i^i x) e. F)
18		fiint 4572	. . . . . . . . . . . . . . . . . 18 |- (A.y e. F A.x e. F (y i^i x) e. F <-> A.y((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
19	17, 18	sylib 198	. . . . . . . . . . . . . . . . 17 |- (F e. Fil -> A.y((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
20	19	19.21bi 1062	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
21	20	imp 350	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ y e. Fin)) -> |^|y e. F)
22		filesn 10545	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> -. (/) e. F)
23	22	adantr 391	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ y e. Fin)) -> -. (/) e. F)
24	12, 21, 23	sylanc 473	. . . . . . . . . . . . . 14 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ y e. Fin)) -> -. |^|y = (/))
25	24	ex 373	. . . . . . . . . . . . 13 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)))
26	25	19.21aiv 1288	. . . . . . . . . . . 12 |- (F e. Fil -> A.y((y (_ F /\ y =/= (/) /\ y e. Fin) -> -. |^|y = (/)))
27	11, 26	syl5 21	. . . . . . . . . . 11 |- (A e. V -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/))))
28	27	com23 32	. . . . . . . . . 10 |- (A e. V -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
29	1, 28	syl 10	. . . . . . . . 9 |- ((A (_ F /\ F e. Fil) -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
30	29	ex 373	. . . . . . . 8 |- (A (_ F -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/)))))
31	30	com14 38	. . . . . . 7 |- (F e. Fil -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (A (_ F -> -. |^|A = (/)))))
32	31	pm2.43i 64	. . . . . 6 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (A (_ F -> -. |^|A = (/))))
33	32	com13 33	. . . . 5 |- (A (_ F -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
34	33	3ad2ant1 802	. . . 4 |- ((A (_ F /\ A =/= (/) /\ A e. Fin) -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/))))
35	34	pm2.43i 64	. . 3 |- ((A (_ F /\ A =/= (/) /\ A e. Fin) -> (F e. Fil -> -. |^|A = (/)))
36	35	com12 11	. 2 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> -. |^|A = (/)))
37		df-ne 1590	. 2 |- (|^|A =/= (/) <-> -. |^|A = (/))
38	36, 37	syl6ibr 213	1 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> |^|A =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777  A.wal 956   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814   i^i cin 2049   (_ wss 2050  (/)c0 2283  U.cuni 2507  |^|cint 2537  Fincfn 4373  Filcfil 10542

This theorem is referenced by:  efilcp 10556  efilcp2 10561  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1o 4139  df-oadd 4141  df-er 4267  df-en 4374  df-fin 4377  df-fil 10543

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