Theorem filint 10562

Description: A filter is closed under taking intersections.

Assertion

Ref	Expression
filint	|- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)

Proof of Theorem filint

Step	Hyp	Ref	Expression
1		eqid 1475	. . . . . . . . 9 |- U.F = U.F
2	1	isfil 10558	. . . . . . . 8 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F)))
3	2	biimpd 153	. . . . . . 7 |- (F e. Fil -> (F e. Fil -> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F)))
4	3	imp 350	. . . . . 6 |- ((F e. Fil /\ F e. Fil) -> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F))
5	4	3simp3d 796	. . . . 5 |- ((F e. Fil /\ F e. Fil) -> A.x e. F A.y e. F (x i^i y) e. F)
6	5	ex 373	. . . 4 |- (F e. Fil -> (F e. Fil -> A.x e. F A.y e. F (x i^i y) e. F))
7		ineq1 2210	. . . . . . 7 |- (x = A -> (x i^i y) = (A i^i y))
8		eleq1 1534	. . . . . . 7 |- ((x i^i y) = (A i^i y) -> ((x i^i y) e. F <-> (A i^i y) e. F))
9	7, 8	syl 10	. . . . . 6 |- (x = A -> ((x i^i y) e. F <-> (A i^i y) e. F))
10		ineq2 2211	. . . . . . 7 |- (y = B -> (A i^i y) = (A i^i B))
11		eleq1 1534	. . . . . . 7 |- ((A i^i y) = (A i^i B) -> ((A i^i y) e. F <-> (A i^i B) e. F))
12	10, 11	syl 10	. . . . . 6 |- (y = B -> ((A i^i y) e. F <-> (A i^i B) e. F))
13	9, 12	rcla42v 1880	. . . . 5 |- ((A e. F /\ B e. F) -> (A.x e. F A.y e. F (x i^i y) e. F -> (A i^i B) e. F))
14	13	com12 11	. . . 4 |- (A.x e. F A.y e. F (x i^i y) e. F -> ((A e. F /\ B e. F) -> (A i^i B) e. F))
15	6, 14	syl6 22	. . 3 |- (F e. Fil -> (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) e. F)))
16	15	pm2.43i 64	. 2 |- (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) e. F))
17	16	3impib 831	1 |- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  (/)c0 2280  U.cuni 2503  Filcfil 10556

This theorem is referenced by:  fipfil 10563  filintf 10569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-fil 10557

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