Theorem fipfil2 10475

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 =/= (/) /\ E.x e. om A ~~ x) -> |^|A =/= (/)))

Distinct variable groups:   x,A   x,F

Proof of Theorem fipfil2

Step	Hyp	Ref	Expression
1		ssexg 2716	. . . . . . . . . 10 |- ((A (_ F /\ F e. Fil) -> A e. V)
2		sseq1 2078	. . . . . . . . . . . . . . 15 |- (y = A -> (y (_ F <-> A (_ F))
3		neeq1 1587	. . . . . . . . . . . . . . 15 |- (y = A -> (y =/= (/) <-> A =/= (/)))
4		breq1 2617	. . . . . . . . . . . . . . . 16 |- (y = A -> (y ~~ x <-> A ~~ x))
5	4	rexbidv 1661	. . . . . . . . . . . . . . 15 |- (y = A -> (E.x e. om y ~~ x <-> E.x e. om A ~~ x))
6	2, 3, 5	3anbi123d 891	. . . . . . . . . . . . . 14 |- (y = A -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) <-> (A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x)))
7		inteq 2531	. . . . . . . . . . . . . . 15 |- (y = A -> |^|y = |^|A)
8		eqeq1 1478	. . . . . . . . . . . . . . . 16 |- (|^|y = |^|A -> (|^|y = (/) <-> |^|A = (/)))
9	8	negbid 610	. . . . . . . . . . . . . . 15 |- (|^|y = |^|A -> (-. |^|y = (/) <-> -. |^|A = (/)))
10	7, 9	syl 10	. . . . . . . . . . . . . 14 |- (y = A -> (-. |^|y = (/) <-> -. |^|A = (/)))
11	6, 10	imbi12d 625	. . . . . . . . . . . . 13 |- (y = A -> (((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> -. |^|y = (/)) <-> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> -. |^|A = (/))))
12	11	cla4gv 1858	. . . . . . . . . . . 12 |- (A e. V -> (A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> -. |^|y = (/)) -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> -. |^|A = (/))))
13		nelneq 1558	. . . . . . . . . . . . . . 15 |- ((|^|y e. F /\ -. (/) e. F) -> -. |^|y = (/))
14		eqid 1473	. . . . . . . . . . . . . . . . . . . . . 22 |- U.F = U.F
15	14	isfil 10469	. . . . . . . . . . . . . . . . . . . . 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)))
16	15	biimpa 416	. . . . . . . . . . . . . . . . . . . 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))
17	16	3simp3d 795	. . . . . . . . . . . . . . . . . . 19 |- ((F e. Fil /\ F e. Fil) -> A.y e. F A.x e. F (y i^i x) e. F)
18	17	anidms 434	. . . . . . . . . . . . . . . . . 18 |- (F e. Fil -> A.y e. F A.x e. F (y i^i x) e. F)
19		fiint 4540	. . . . . . . . . . . . . . . . . 18 |- (A.y e. F A.x e. F (y i^i x) e. F <-> A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
20	18, 19	sylib 198	. . . . . . . . . . . . . . . . 17 |- (F e. Fil -> A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
21	20	19.21bi 1058	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
22	21	imp 350	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x)) -> |^|y e. F)
23		filesn 10470	. . . . . . . . . . . . . . . 16 |- (F e. Fil -> -. (/) e. F)
24	23	adantr 389	. . . . . . . . . . . . . . 15 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x)) -> -. (/) e. F)
25	13, 22, 24	sylanc 471	. . . . . . . . . . . . . 14 |- ((F e. Fil /\ (y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x)) -> -. |^|y = (/))
26	25	ex 373	. . . . . . . . . . . . 13 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> -. |^|y = (/)))
27	26	19.21aiv 1284	. . . . . . . . . . . 12 |- (F e. Fil -> A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> -. |^|y = (/)))
28	12, 27	syl5 21	. . . . . . . . . . 11 |- (A e. V -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> -. |^|A = (/))))
29	28	com23 32	. . . . . . . . . 10 |- (A e. V -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/))))
30	1, 29	syl 10	. . . . . . . . 9 |- ((A (_ F /\ F e. Fil) -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/))))
31	30	ex 373	. . . . . . . 8 |- (A (_ F -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/)))))
32	31	com14 38	. . . . . . 7 |- (F e. Fil -> (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (A (_ F -> -. |^|A = (/)))))
33	32	pm2.43i 64	. . . . . 6 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (A (_ F -> -. |^|A = (/))))
34	33	com13 33	. . . . 5 |- (A (_ F -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/))))
35	34	3ad2ant1 799	. . . 4 |- ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/))))
36	35	pm2.43i 64	. . 3 |- ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> (F e. Fil -> -. |^|A = (/)))
37	36	com12 11	. 2 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> -. |^|A = (/)))
38		df-ne 1584	. 2 |- (|^|A =/= (/) <-> -. |^|A = (/))
39	37, 38	syl6ibr 213	1 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> |^|A =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774  A.wal 952   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807   i^i cin 2042   (_ wss 2043  (/)c0 2276  U.cuni 2498  |^|cint 2528   class class class wbr 2614  omcom 3126   ~~ cen 4354  Filcfil 10467

This theorem is referenced by:  efilcp 10481  efilcp2 10486  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1o 4123  df-oadd 4125  df-er 4251  df-en 4357  df-fil 10468

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