Nuprl Lemma : length-one-member
∀[T:Type]. ∀[L:T List]. ∀[x,y:T]. (x = y ∈ T) supposing ((y ∈ L) and (x ∈ L)) supposing ||L|| = 1 ∈ ℤ
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
false: False,
prop: ℙ,
cons: [a / b],
top: Top,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
iff: P ⇐⇒ Q
Lemmas referenced :
member_singleton,
list_wf,
length_wf,
cons_wf,
l_member_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
non_neg_length,
length_of_cons_lemma,
product_subtype_list,
equal_wf,
int_subtype_base,
subtype_base_sq,
length_of_nil_lemma,
list-cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
isect_memberEquality,
axiomEquality,
because_Cache,
hypothesis_subsumption,
productElimination,
voidEquality,
rename,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
independent_pairFormation,
computeAll,
addEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[x,y:T]. (x = y) supposing ((y \mmember{} L) and (x \mmember{} L)) supposing ||L|| = 1
Date html generated:
2016_05_14-PM-01_27_50
Last ObjectModification:
2016_01_15-AM-08_28_54
Theory : list_1
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