Theorem fipfil 10549

Description: The intersection of two elements of a filter can't be empty.

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		nelneq 1564	. . . 4 |- (((A i^i B) e. F /\ -. (/) e. F) -> -. (A i^i B) = (/))
2		filint 10548	. . . . 5 |- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)
3	2	3expb 836	. . . 4 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> (A i^i B) e. F)
4		filesn 10545	. . . . 5 |- (F e. Fil -> -. (/) e. F)
5	4	adantr 391	. . . 4 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> -. (/) e. F)
6	1, 3, 5	sylanc 473	. . 3 |- ((F e. Fil /\ (A e. F /\ B e. F)) -> -. (A i^i B) = (/))
7	6	ex 373	. 2 |- (F e. Fil -> ((A e. F /\ B e. F) -> -. (A i^i B) = (/)))
8		df-ne 1590	. 2 |- ((A i^i B) =/= (/) <-> -. (A i^i B) = (/))
9	7, 8	syl6ibr 213	1 |- (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588   i^i cin 2049  (/)c0 2283  Filcfil 10542

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-uni 2508  df-fil 10543

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