Theorem isufil 16388

Description: The property of being an ultrafilter.

Hypothesis

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

Assertion

Ref	Expression
isufil	|- (F e. UFil <-> (F e. Fil /\ A.x e. ~P X(x e. F \/ (X \ x) e. F)))

Distinct variable groups:   x,F   x,X

Proof of Theorem isufil

Step	Hyp	Ref	Expression
1		unieq 3375	. . . . 5 |- (f = F -> U.f = U.F)
2		isufil.1	. . . . 5 |- X = U.F
3	1, 2	syl6eqr 2195	. . . 4 |- (f = F -> U.f = X)
4		pweq 3230	. . . 4 |- (U.f = X -> ~PU.f = ~PX)
5	3, 4	syl 13	. . 3 |- (f = F -> ~PU.f = ~PX)
6		eleq2 2205	. . . 4 |- (f = F -> (x e. f <-> x e. F))
7	3	difeq1d 2945	. . . . 5 |- (f = F -> (U.f \ x) = (X \ x))
8		id 15	. . . . 5 |- (f = F -> f = F)
9	7, 8	eleq12d 2212	. . . 4 |- (f = F -> ((U.f \ x) e. f <-> (X \ x) e. F))
10	6, 9	orbi12d 762	. . 3 |- (f = F -> ((x e. f \/ (U.f \ x) e. f) <-> (x e. F \/ (X \ x) e. F)))
11	5, 10	raleqbidv 2520	. 2 |- (f = F -> (A.x e. ~P U.f(x e. f \/ (U.f \ x) e. f) <-> A.x e. ~P X(x e. F \/ (X \ x) e. F)))
12		df-ufil 16387	. 2 |- UFil = {f e. Fil | A.x e. ~P U.f(x e. f \/ (U.f \ x) e. f)}
13	11, 12	elrab2 2656	1 |- (F e. UFil <-> (F e. Fil /\ A.x e. ~P X(x e. F \/ (X \ x) e. F)))

Syntax hints:   <-> wb 219   \/ wo 336   /\ wa 337   = wceq 1586   e. wcel 1588  A.wral 2355   \ cdif 2824  ~Pcpw 3227  U.cuni 3366  Filcfil 11102  UFilcufil 16386

This theorem is referenced by:  isufil2 16389  ufilfil 16390  ufilss 16391  ufileu 16397  fixufil 16400

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-ral 2359  df-rex 2360  df-rab 2362  df-v 2540  df-dif 2830  df-in 2834  df-ss 2836  df-pw 3229  df-uni 3367  df-ufil 16387

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