Nuprl Lemma : l_all_single
∀[T:Type]. ∀t:T. ∀[P:{x:T| x = t ∈ T} ⟶ ℙ]. ((∀x∈[t].P[x]) ⇐⇒ P[t])
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
cons: [a / b],
nil: [],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
all_wf,
int_seg_wf,
equal_wf,
select_wf,
cons_wf,
nil_wf,
sq_stable__le,
length-singleton,
select-cons-hd,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
zero-add,
add_functionality_wrt_le,
add-associates,
le-add-cancel2,
lelt_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,
isectElimination,
natural_numberEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
setEquality,
because_Cache,
dependent_set_memberEquality,
cumulativity,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
addEquality,
minusEquality,
intEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}t:T. \mforall{}[P:\{x:T| x = t\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x\mmember{}[t].P[x]) \mLeftarrow{}{}\mRightarrow{} P[t])
Date html generated:
2017_04_14-AM-08_40_06
Last ObjectModification:
2017_02_27-PM-03_30_52
Theory : list_0
Home
Index