Theorem filusb 10530

Description: The underlying set belongs to the filter.

Hypothesis

Ref	Expression
filusb.1	|- X = U.F

Assertion

Ref	Expression
filusb	|- (F e. Fil -> X e. F)

Proof of Theorem filusb

Step	Hyp	Ref	Expression
1		filusb.1	. . . . 5 |- X = U.F
2	1	isfil 10527	. . . 4 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ X e. F) /\ A.xA.y((x e. F /\ y (_ X /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F)))
3	2	ibi 591	. . 3 |- (F e. Fil -> ((-. (/) e. F /\ X e. F) /\ A.xA.y((x e. F /\ y (_ X /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F))
4	3	3simp1d 793	. 2 |- (F e. Fil -> (-. (/) e. F /\ X e. F))
5	4	pm3.27d 325	1 |- (F e. Fil -> X e. F)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957  A.wral 1644   i^i cin 2044   (_ wss 2045  (/)c0 2278  U.cuni 2500  Filcfil 10525

This theorem is referenced by:  emnfil 10534  filintf 10537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-v 1810  df-in 2049  df-ss 2051  df-uni 2501  df-fil 10526

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