Theorem fipfil 10271

Description: The intersection of two elements of a filter can't be empty. (Contributed by FL, 19-Sep-2007.)

Assertion

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

Proof of Theorem fipfil

Step	Hyp	Ref	Expression
1		filint 10269	. . . . 5 |- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)
2	1	3expb 1068	. . . 4 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> (A i^i B) e. F)
3		filesn 10268	. . . . 5 |- (F e. Fil -> -. (/) e. F)
4	3	adantr 425	. . . 4 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> -. (/) e. F)
5		nelneq 1985	. . . 4 |- (((A i^i B) e. F /\ -. (/) e. F) -> -. (A i^i B) = (/))
6	2, 4, 5	syl11anc 524	. . 3 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> -. (A i^i B) = (/))
7	6	ex 402	. 2 |- (F e. Fil -> ((A e. F /\ B e. F) -> -. (A i^i B) = (/)))
8		df-ne 2019	. 2 |- ((A i^i B) =/= (/) <-> -. (A i^i B) = (/))
9	7, 8	syl6ibr 230	1 |- (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   i^i cin 2592  (/)c0 2875  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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178  df-fil 10265

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