Theorem efilcp 14922

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.

Hypothesis

Ref	Expression
efilcp.1	|- B = {z | E.y(y C_ A /\ y e. Fin /\ z = |^|y)}

Assertion

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

Distinct variable groups:   x,A,y,z   x,B,y,z   x,X,y,z

Proof of Theorem efilcp

Step	Hyp	Ref	Expression
1		efilcp.1	. . . . . 6 |- B = {z | E.y(y C_ A /\ y e. Fin /\ z = |^|y)}
2	1	fgsb 14921	. . . . 5 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. B -> {z e. ~PX | E.y e. B y C_ z} e. Fil))
3	2	imp 377	. . . 4 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. B) -> {z e. ~PX | E.y e. B y C_ z} e. Fil)
4		simpl1 879	. . . . . 6 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. B) -> A C_ ~PX)
5		abfi 10215	. . . . . . . . 9 |- A C_ {z | E.y(y C_ A /\ y e. Fin /\ z = |^|y)}
6	5, 1	sseqtr4i 2650	. . . . . . . 8 |- A C_ B
7		ssel2 2616	. . . . . . . . 9 |- ((A C_ B /\ z e. A) -> z e. B)
8		ssid 2634	. . . . . . . . . 10 |- z C_ z
9		sseq1 2637	. . . . . . . . . . 11 |- (y = z -> (y C_ z <-> z C_ z))
10	9	rcla4ev 2381	. . . . . . . . . 10 |- ((z e. B /\ z C_ z) -> E.y e. B y C_ z)
11	8, 10	mpan2 760	. . . . . . . . 9 |- (z e. B -> E.y e. B y C_ z)
12	7, 11	syl 12	. . . . . . . 8 |- ((A C_ B /\ z e. A) -> E.y e. B y C_ z)
13	6, 12	mpan 759	. . . . . . 7 |- (z e. A -> E.y e. B y C_ z)
14	13	rgen 2159	. . . . . 6 |- A.z e. A E.y e. B y C_ z
15	4, 14	jctir 317	. . . . 5 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. B) -> (A C_ ~PX /\ A.z e. A E.y e. B y C_ z))
16		ssrab 2685	. . . . 5 |- (A C_ {z e. ~PX | E.y e. B y C_ z} <-> (A C_ ~PX /\ A.z e. A E.y e. B y C_ z))
17	15, 16	sylibr 217	. . . 4 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. B) -> A C_ {z e. ~PX | E.y e. B y C_ z})
18		sseq2 2639	. . . . 5 |- (x = {z e. ~PX | E.y e. B y C_ z} -> (A C_ x <-> A C_ {z e. ~PX | E.y e. B y C_ z}))
19	18	rcla4ev 2381	. . . 4 |- (({z e. ~PX | E.y e. B y C_ z} e. Fil /\ A C_ {z e. ~PX | E.y e. B y C_ z}) -> E.x e. Fil A C_ x)
20	3, 17, 19	syl11anc 524	. . 3 |- (((A C_ ~PX /\ X e. _V /\ A =/= (/)) /\ -. (/) e. B) -> E.x e. Fil A C_ x)
21	20	ex 402	. 2 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. B -> E.x e. Fil A C_ x))
22		sstr 2625	. . . . . . . . . . . . . . . . . . 19 |- ((y C_ A /\ A C_ x) -> y C_ x)
23		df-ne 2019	. . . . . . . . . . . . . . . . . . . . 21 |- (y =/= (/) <-> -. y = (/))
24		fipfil2 10272	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. Fil -> ((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)))
25		pm2.27 76	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((y C_ x /\ y =/= (/) /\ y e. Fin) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> |^|y =/= (/)))
26		necom 2094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((/) =/= |^|y <-> |^|y =/= (/))
27		df-ne 2019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((/) =/= |^|y <-> -. (/) = |^|y)
28	27	biimpi 168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((/) =/= |^|y -> -. (/) = |^|y)
29	26, 28	sylbir 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (|^|y =/= (/) -> -. (/) = |^|y)
30	25, 29	syl6 25	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((y C_ x /\ y =/= (/) /\ y e. Fin) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> -. (/) = |^|y))
31	30	3exp 1066	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y C_ x -> (y =/= (/) -> (y e. Fin -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> -. (/) = |^|y))))
32	31	com34 40	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y C_ x -> (y =/= (/) -> (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> (y e. Fin -> -. (/) = |^|y))))
33	32	com4t 44	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y =/= (/)) -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
34	24, 33	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. Fil -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
35	34	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (y e. Fin -> (y C_ x -> (y =/= (/) -> -. (/) = |^|y))))
36	35	com14 42	. . . . . . . . . . . . . . . . . . . . 21 |- (y =/= (/) -> (y e. Fin -> (y C_ x -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
37	23, 36	sylbir 218	. . . . . . . . . . . . . . . . . . . 20 |- (-. y = (/) -> (y e. Fin -> (y C_ x -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
38	37	com13 37	. . . . . . . . . . . . . . . . . . 19 |- (y C_ x -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
39	22, 38	syl 12	. . . . . . . . . . . . . . . . . 18 |- ((y C_ A /\ A C_ x) -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
40	39	ex 402	. . . . . . . . . . . . . . . . 17 |- (y C_ A -> (A C_ x -> (y e. Fin -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y)))))
41	40	com23 36	. . . . . . . . . . . . . . . 16 |- (y C_ A -> (y e. Fin -> (A C_ x -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y)))))
42	41	imp 377	. . . . . . . . . . . . . . 15 |- ((y C_ A /\ y e. Fin) -> (A C_ x -> (-. y = (/) -> ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) = |^|y))))
43	42	com14 42	. . . . . . . . . . . . . 14 |- ((x e. Fil /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (A C_ x -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))))
44	43	ex 402	. . . . . . . . . . . . 13 |- (x e. Fil -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (A C_ x -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))))
45	44	com23 36	. . . . . . . . . . . 12 |- (x e. Fil -> (A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))))
46	45	imp31 389	. . . . . . . . . . 11 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> (-. y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y)))
47		inteq 3217	. . . . . . . . . . . . . . . 16 |- ((/) = y -> |^|(/) = |^|y)
48		int0 3230	. . . . . . . . . . . . . . . . . 18 |- |^|(/) = _V
49	48	eqeq1i 1891	. . . . . . . . . . . . . . . . 17 |- (|^|(/) = |^|y <-> _V = |^|y)
50		vn0 2882	. . . . . . . . . . . . . . . . . 18 |- _V =/= (/)
51		neeq1 2024	. . . . . . . . . . . . . . . . . 18 |- (_V = |^|y -> (_V =/= (/) <-> |^|y =/= (/)))
52	50, 51	mpbii 210	. . . . . . . . . . . . . . . . 17 |- (_V = |^|y -> |^|y =/= (/))
53	49, 52	sylbi 216	. . . . . . . . . . . . . . . 16 |- (|^|(/) = |^|y -> |^|y =/= (/))
54	47, 53	syl 12	. . . . . . . . . . . . . . 15 |- ((/) = y -> |^|y =/= (/))
55	54	eqcoms 1887	. . . . . . . . . . . . . 14 |- (y = (/) -> |^|y =/= (/))
56	55	necomd 2095	. . . . . . . . . . . . 13 |- (y = (/) -> (/) =/= |^|y)
57	56, 27	sylib 215	. . . . . . . . . . . 12 |- (y = (/) -> -. (/) = |^|y)
58	57	a1d 15	. . . . . . . . . . 11 |- (y = (/) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))
59	46, 58	pm2.61d2 143	. . . . . . . . . 10 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> ((y C_ A /\ y e. Fin) -> -. (/) = |^|y))
60	59	imp 377	. . . . . . . . 9 |- ((((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) /\ (y C_ A /\ y e. Fin)) -> -. (/) = |^|y)
61		nan 753	. . . . . . . . 9 |- ((((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))
62	60, 61	mpbir 207	. . . . . . . 8 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
63		df-3an 860	. . . . . . . . 9 |- ((y C_ A /\ y e. Fin /\ (/) = |^|y) <-> ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
64	63	notbii 204	. . . . . . . 8 |- (-. (y C_ A /\ y e. Fin /\ (/) = |^|y) <-> -. ((y C_ A /\ y e. Fin) /\ (/) = |^|y))
65	62, 64	sylibr 217	. . . . . . 7 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (y C_ A /\ y e. Fin /\ (/) = |^|y))
66	65	nexdv 1711	. . . . . 6 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. E.y(y C_ A /\ y e. Fin /\ (/) = |^|y))
67		0ex 3446	. . . . . . . 8 |- (/) e. _V
68		eqeq1 1890	. . . . . . . . . 10 |- (z = (/) -> (z = |^|y <-> (/) = |^|y))
69	68	3anbi3d 1174	. . . . . . . . 9 |- (z = (/) -> ((y C_ A /\ y e. Fin /\ z = |^|y) <-> (y C_ A /\ y e. Fin /\ (/) = |^|y)))
70	69	exbidv 1657	. . . . . . . 8 |- (z = (/) -> (E.y(y C_ A /\ y e. Fin /\ z = |^|y) <-> E.y(y C_ A /\ y e. Fin /\ (/) = |^|y)))
71	67, 70, 1	elab2 2407	. . . . . . 7 |- ((/) e. B <-> E.y(y C_ A /\ y e. Fin /\ (/) = |^|y))
72	71	notbii 204	. . . . . 6 |- (-. (/) e. B <-> -. E.y(y C_ A /\ y e. Fin /\ (/) = |^|y))
73	66, 72	sylibr 217	. . . . 5 |- (((x e. Fil /\ A C_ x) /\ (A C_ ~PX /\ X e. _V /\ A =/= (/))) -> -. (/) e. B)
74	73	exp31 407	. . . 4 |- (x e. Fil -> (A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> -. (/) e. B)))
75	74	r19.23aiv 2211	. . 3 |- (E.x e. Fil A C_ x -> ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> -. (/) e. B))
76	75	com12 14	. 2 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (E.x e. Fil A C_ x -> -. (/) e. B))
77	21, 76	impbid 574	1 |- ((A C_ ~PX /\ X e. _V /\ A =/= (/)) -> (-. (/) e. B <-> 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  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  |^|cint 3214  Fincfn 5426  Filcfil 10264

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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