Nuprl Lemma : member-fset-remove
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[x,y:T]. uiff(x ∈ fset-remove(eq;y;s);x ∈ s ∧ (¬(x = y ∈ T)))
Proof
Definitions occuring in Statement :
fset-remove: fset-remove(eq;y;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fset-remove: fset-remove(eq;y;s),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
deq: EqDecider(T),
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqof: eqof(d)
Lemmas referenced :
equal_wf,
fset-member_witness,
fset-member_wf,
not_wf,
assert_wf,
bnot_wf,
eqof_wf,
uiff_wf,
fset-filter_wf,
fset-remove_wf,
fset_wf,
deq_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
assert_witness,
member-fset-filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lambdaFormation,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
because_Cache,
lambdaEquality,
dependent_functionElimination,
productEquality,
applyEquality,
setElimination,
rename,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
independent_isectElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[x,y:T].
uiff(x \mmember{} fset-remove(eq;y;s);x \mmember{} s \mwedge{} (\mneg{}(x = y)))
Date html generated:
2017_04_17-AM-09_19_42
Last ObjectModification:
2017_02_27-PM-05_22_58
Theory : finite!sets
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