Nuprl Lemma : assert-fset-antichain
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac:fset(fset(T))].
uiff(↑fset-antichain(eq;ac);∀xs,ys:fset(T). (¬xs ⊆≠ ys) supposing (xs ∈ ac and ys ∈ ac))
Proof
Definitions occuring in Statement :
fset-antichain: fset-antichain(eq;ac),
f-proper-subset: xs ⊆≠ ys,
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
fset-antichain: fset-antichain(eq;ac),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
assert-fset-pairwise,
fset_wf,
deq-fset_wf,
iff_weakening_uiff,
assert_wf,
fset-pairwise_wf,
bnot_wf,
f-proper-subset-dec_wf,
all_wf,
isect_wf,
fset-member_wf,
f-proper-subset_wf,
assert_witness,
not_wf,
uiff_wf,
fset-antichain_wf,
deq_wf,
iff_transitivity,
assert_of_bnot,
assert-f-proper-subset-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
addLevel,
productElimination,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
universeEquality,
lambdaFormation,
independent_pairEquality,
allFunctionality,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[ac:fset(fset(T))].
uiff(\muparrow{}fset-antichain(eq;ac);\mforall{}xs,ys:fset(T). (\mneg{}xs \msubseteq{}\mneq{} ys) supposing (xs \mmember{} ac and ys \mmember{} ac))
Date html generated:
2016_05_14-PM-03_42_43
Last ObjectModification:
2015_12_26-PM-06_39_59
Theory : finite!sets
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