Nuprl Lemma : fset-ac-glb_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac1,ac2:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ].
(fset-ac-glb(eq;ac1;ac2) ∈ {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} )
Proof
Definitions occuring in Statement :
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ
Lemmas referenced :
fset-minimals-antichain,
f-union_wf,
fset_wf,
deq-fset_wf,
fset-image_wf,
fset-union_wf,
fset-minimals_wf,
f-proper-subset-dec_wf,
assert_wf,
fset-antichain_wf,
set_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
lambdaEquality,
dependent_set_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[ac1,ac2:\{ac:fset(fset(T))| \muparrow{}fset-antichain(eq;ac)\} ].
(fset-ac-glb(eq;ac1;ac2) \mmember{} \{ac:fset(fset(T))| \muparrow{}fset-antichain(eq;ac)\} )
Date html generated:
2016_05_14-PM-03_49_17
Last ObjectModification:
2015_12_26-PM-06_36_14
Theory : finite!sets
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