Nuprl Lemma : member_singleton
∀[T:Type]. ∀a,x:T. ((x ∈ [a]) ⇐⇒ x = a ∈ T)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
cons: [a / b],
nil: [],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
subtract: n - m,
less_than: a < b
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
exists_wf,
nat_wf,
less_than_wf,
equal_wf,
select_wf,
cons_wf,
nil_wf,
sq_stable__le,
length-singleton,
decidable__assert,
eq_int_wf,
assert_of_eq_int,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
neg_assert_of_eq_int,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less-iff-le,
le-add-cancel2,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
productElimination,
isectElimination,
lambdaEquality,
productEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
cumulativity,
hypothesisEquality,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
unionElimination,
instantiate,
intEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
applyEquality,
minusEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}a,x:T. ((x \mmember{} [a]) \mLeftarrow{}{}\mRightarrow{} x = a)
Date html generated:
2017_04_14-AM-08_37_24
Last ObjectModification:
2017_02_27-PM-03_29_24
Theory : list_0
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