Nuprl Lemma : fset-extensionality
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:fset(T)]. uiff(x = y ∈ fset(T);∀[a:T]. uiff(a ∈ x;a ∈ y))
Proof
Definitions occuring in Statement :
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
fset: fset(T),
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
fset-member: a ∈ s,
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
equiv_rel: EquivRel(T;x,y.E[x; y])
Lemmas referenced :
and_wf,
equal_wf,
fset_wf,
fset-member_wf,
fset-member_witness,
uall_wf,
uiff_wf,
deq_wf,
set-equal-equiv,
list_wf,
list_subtype_fset,
set-equal_wf,
equal-wf-base,
quotient-member-eq,
l_member_wf,
assert-deq-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
hyp_replacement,
thin,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
productElimination,
independent_functionElimination,
cumulativity,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
lambdaEquality,
axiomEquality,
universeEquality,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
lambdaFormation,
applyEquality,
independent_isectElimination,
comment,
dependent_functionElimination,
productEquality,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:fset(T)]. uiff(x = y;\mforall{}[a:T]. uiff(a \mmember{} x;a \mmember{} y))
Date html generated:
2017_04_17-AM-09_18_50
Last ObjectModification:
2017_02_27-PM-05_22_26
Theory : finite!sets
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