Theorem efilcp2 14926

Description: A filter containing a set A exists iff A has the finite intersection property (i.e. no finite intersection of elements of A is empty). Bourbaki TG I.37 prop. 1.

Assertion

Ref	Expression
efilcp2	|- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. ( fi ` A) <-> E.x e. Fil A C_ x))

Distinct variable groups:   x,A   x,X

Proof of Theorem efilcp2

Step	Hyp	Ref	Expression
1		fgsb2 14925	. . . . 5 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. ( fi ` A) -> {z e. ~PX | E.y e. ( fi ` A)y C_ z} e. Fil))
2	1	imp 377	. . . 4 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> {z e. ~PX | E.y e. ( fi ` A)y C_ z} e. Fil)
3		simpl1 879	. . . . . 6 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> A C_ ~PX)
4		pwexg 3489	. . . . . . . . . . . 12 |- (X e. _V -> ~PX e. _V)
5		elpw2g 3463	. . . . . . . . . . . 12 |- (~PX e. _V -> (A e. ~P~PX <-> A C_ ~PX))
6	4, 5	syl 12	. . . . . . . . . . 11 |- (X e. _V -> (A e. ~P~PX <-> A C_ ~PX))
7	6	biimprd 171	. . . . . . . . . 10 |- (X e. _V -> (A C_ ~PX -> A e. ~P~PX))
8		elisset 2299	. . . . . . . . . . 11 |- (A e. ~P~PX -> A e. _V)
9	8	a1d 15	. . . . . . . . . 10 |- (A e. ~P~PX -> (A =/= (/) -> A e. _V))
10	7, 9	syl6com 64	. . . . . . . . 9 |- (A C_ ~PX -> (X e. _V -> (A =/= (/) -> A e. _V)))
11	10	3imp 1061	. . . . . . . 8 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> A e. _V)
12	11	adantr 425	. . . . . . 7 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> A e. _V)
13		abfi2 10216	. . . . . . . 8 |- (A e. _V -> A C_ ( fi ` A))
14		ssel2 2616	. . . . . . . . . 10 |- ((A C_ ( fi ` A) /\ z e. A) -> z e. ( fi ` A))
15		ssid 2634	. . . . . . . . . . 11 |- z C_ z
16		sseq1 2637	. . . . . . . . . . . 12 |- (y = z -> (y C_ z <-> z C_ z))
17	16	rcla4ev 2381	. . . . . . . . . . 11 |- ((z e. ( fi ` A) /\ z C_ z) -> E.y e. ( fi ` A)y C_ z)
18	15, 17	mpan2 760	. . . . . . . . . 10 |- (z e. ( fi ` A) -> E.y e. ( fi ` A)y C_ z)
19	14, 18	syl 12	. . . . . . . . 9 |- ((A C_ ( fi ` A) /\ z e. A) -> E.y e. ( fi ` A)y C_ z)
20	19	r19.21aiva 2176	. . . . . . . 8 |- (A C_ ( fi ` A) -> A.z e. A E.y e. ( fi ` A)y C_ z)
21	13, 20	syl 12	. . . . . . 7 |- (A e. _V -> A.z e. A E.y e. ( fi ` A)y C_ z)
22	12, 21	syl 12	. . . . . 6 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> A.z e. A E.y e. ( fi ` A)y C_ z)
23	3, 22	jca 310	. . . . 5 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> (A C_ ~PX /\ A.z e. A E.y e. ( fi ` A)y C_ z))
24		ssrab 2685	. . . . 5 |- (A C_ {z e. ~PX | E.y e. ( fi ` A)y C_ z} <-> (A C_ ~PX /\ A.z e. A E.y e. ( fi ` A)y C_ z))
25	23, 24	sylibr 217	. . . 4 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> A C_ {z e. ~PX | E.y e. ( fi ` A)y C_ z})
26		sseq2 2639	. . . . 5 |- (x = {z e. ~PX | E.y e. ( fi ` A)y C_ z} -> (A C_ x <-> A C_ {z e. ~PX | E.y e. ( fi ` A)y C_ z}))
27	26	rcla4ev 2381	. . . 4 |- (({z e. ~PX | E.y e. ( fi ` A)y C_ z} e. Fil /\ A C_ {z e. ~PX | E.y e. ( fi ` A)y C_ z}) -> E.x e. Fil A C_ x)
28	2, 25, 27	syl11anc 524	. . 3 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. ( fi ` A)) -> E.x e. Fil A C_ x)
29	28	ex 402	. 2 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. ( fi ` A) -> E.x e. Fil A C_ x))
30		0ex 3446	. . . . . . . . . . 11 |- (/) e. _V
31		sppfi 10218	. . . . . . . . . . 11 |- (((/) e. _V /\ A e. _V) -> ((/) e. ( fi ` A) <-> E.y(y C_ A /\ y e. Fin /\ (/) = |^|y)))
32	30, 31	mpan 759	. . . . . . . . . 10 |- (A e. _V -> ((/) e. ( fi ` A) <-> E.y(y C_ A /\ y e. Fin /\ (/) = |^|y)))
33	32	notbid 673	. . . . . . . . 9 |- (A e. _V -> (-. (/) e. ( fi ` A) <-> -. E.y(y C_ A /\ y e. Fin /\ (/) = |^|y)))
34		sstr 2625	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y C_ A /\ A C_ x) -> y C_ x)
35		df-ne 2019	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y =/= (/) <-> -. y = (/))
36		fipfil2 10272	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. Fil -> ((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)))
37		pm2.27 76	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((y C_ x /\ y =/= (/) /\ y e. Fin) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> |^|y =/= (/)))
38		necom 2094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((/) =/= |^|y <-> |^|y =/= (/))
39		df-ne 2019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((/) =/= |^|y <-> -. (/) = |^|y)
40	39	biimpi 168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((/) =/= |^|y -> -. (/) = |^|y)
41	38, 40	sylbir 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (|^|y =/= (/) -> -. (/) = |^|y)
42	37, 41	syl6 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((y C_ x /\ y =/= (/) /\ y e. Fin) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> -. (/) = |^|y))
43	42	3exp 1066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (y C_ x -> (y =/= (/) -> (y e. Fin -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> -. (/) = |^|y))))
44	43	com34 40	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (y C_ x -> (y =/= (/) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> (y e. Fin -> -. (/) = |^|y))))
45	44	com4t 44	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
46	36, 45	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. Fil -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
47	46	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
48	47	com14 42	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y =/= (/) -> (y e. Fin -> (y C_ x -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
49	35, 48	sylbir 218	. . . . . . . . . . . . . . . . . . . . . . 23 |- (-. y = (/) -> (y e. Fin -> (y C_ x -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
50	49	com13 37	. . . . . . . . . . . . . . . . . . . . . 22 |- (y C_ x -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
51	34, 50	syl 12	. . . . . . . . . . . . . . . . . . . . 21 |- ((y C_ A /\ A C_ x) -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
52	51	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- (y C_ A -> (A C_ x -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y)))))
53	52	com23 36	. . . . . . . . . . . . . . . . . . 19 |- (y C_ A -> (y e. Fin -> (A C_ x -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y)))))
54	53	imp 377	. . . . . . . . . . . . . . . . . 18 |- ((y C_ A /\ y e. Fin) -> (A C_ x -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
55	54	com14 42	. . . . . . . . . . . . . . . . 17 |- ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (A C_ x -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))))
56	55	ex 402	. . . . . . . . . . . . . . . 16 |- (x e. Fil -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (A C_ x -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))))
57	56	com23 36	. . . . . . . . . . . . . . 15 |- (x e. Fil -> (A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))))
58	57	3imp 1061	. . . . . . . . . . . . . 14 |- ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))
59		inteq 3217	. . . . . . . . . . . . . . . . . . 19 |- ((/) = y -> |^|(/) = |^|y)
60		int0 3230	. . . . . . . . . . . . . . . . . . . . 21 |- |^|(/) = _V
61	60	eqeq1i 1891	. . . . . . . . . . . . . . . . . . . 20 |- (|^|(/) = |^|y <-> _V = |^|y)
62		vn0 2882	. . . . . . . . . . . . . . . . . . . . 21 |- _V =/= (/)
63		neeq1 2024	. . . . . . . . . . . . . . . . . . . . 21 |- (_V = |^|y -> (_V =/= (/) <-> |^|y =/= (/)))
64	62, 63	mpbii 210	. . . . . . . . . . . . . . . . . . . 20 |- (_V = |^|y -> |^|y =/= (/))
65	61, 64	sylbi 216	. . . . . . . . . . . . . . . . . . 19 |- (|^|(/) = |^|y -> |^|y =/= (/))
66	59, 65	syl 12	. . . . . . . . . . . . . . . . . 18 |- ((/) = y -> |^|y =/= (/))
67	66	eqcoms 1887	. . . . . . . . . . . . . . . . 17 |- (y = (/) -> |^|y =/= (/))
68	67	necomd 2095	. . . . . . . . . . . . . . . 16 |- (y = (/) -> (/) =/= |^|y)
69	68, 39	sylib 215	. . . . . . . . . . . . . . 15 |- (y = (/) -> -. (/) = |^|y)
70	69	a1d 15	. . . . . . . . . . . . . 14 |- (y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))
71	58, 70	pm2.61d2 143	. . . . . . . . . . . . 13 |- ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))
72	71	imp 377	. . . . . . . . . . . 12 |- (((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) /\ (y C_ A /\ y e. Fin)) -> -. (/) = |^|y)
73		nan 753	. . . . . . . . . . . 12 |- (((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. ((y C_ A /\ y e. Fin) /\ (/) = |^|y)) <-> (((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) /\ (y C_ A /\ y e. Fin)) -> -. (/) = |^|y))
74	72, 73	mpbir 207	. . . . . . . . . . 11 |- ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
75		df-3an 860	. . . . . . . . . . . 12 |- ((y C_ A /\ y e. Fin /\ (/) = |^|y) <-> ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
76	75	notbii 204	. . . . . . . . . . 11 |- (-. (y C_ A /\ y e. Fin /\ (/) = |^|y) <-> -. ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
77	74, 76	sylibr 217	. . . . . . . . . 10 |- ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (y C_ A /\ y e. Fin /\ (/) = |^|y))
78	77	nexdv 1711	. . . . . . . . 9 |- ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. E.y(y C_ A /\ y e. Fin /\ (/) = |^|y))
79	33, 78	syl5bir 227	. . . . . . . 8 |- (A e. _V -> ((x e. Fil /\ A C_ x /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) e. ( fi ` A)))
80	79	3expd 1085	. . . . . . 7 |- (A e. _V -> (x e. Fil -> (A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> -. (/) e. ( fi ` A)))))
81	80	com24 41	. . . . . 6 |- (A e. _V -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (A C_ x -> (x e. Fil -> -. (/) e. ( fi ` A)))))
82	11, 81	mpcom 60	. . . . 5 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (A C_ x -> (x e. Fil -> -. (/) e. ( fi ` A))))
83	82	com13 37	. . . 4 |- (x e. Fil -> (A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> -. (/) e. ( fi ` A))))
84	83	r19.23aiv 2211	. . 3 |- (E.x e. Fil A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> -. (/) e. ( fi ` A)))
85	84	com12 14	. 2 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (E.x e. Fil A C_ x -> -. (/) e. ( fi ` A)))
86	29, 85	impbid 574	1 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. ( fi ` A) <-> E.x e. Fil A C_ x))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  |^|cint 3214  ` cfv 3998  Fincfn 5426   fi cfi 10210  Filcfil 10264

This theorem is referenced by:  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-fi 10211  df-fil 10265

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