Nuprl Lemma : list-cardinality-le
∀[T:Type]. ∀L:T List. ((∀x:T. (x ∈ L)) ⇒ |T| ≤ ||L||)
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
l_member: (x ∈ l),
length: ||as||,
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
cardinality-le: |T| ≤ n,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
surject: Surj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
ge: i ≥ j
Lemmas referenced :
nat_properties,
equal_wf,
lelt_wf,
list_wf,
l_member_wf,
all_wf,
surject_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
imageElimination,
universeEquality,
dependent_set_memberEquality,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. ((\mforall{}x:T. (x \mmember{} L)) {}\mRightarrow{} |T| \mleq{} ||L||)
Date html generated:
2016_05_14-PM-01_51_46
Last ObjectModification:
2016_01_15-AM-08_15_56
Theory : list_1
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