Nuprl Lemma : setmem-natset
∀n:ℕ. ∀x:Set{i:l}. ((x ∈ natset(n)) ⇐⇒ ∃i:ℕn. seteq(x;natset(i)))
Proof
Definitions occuring in Statement :
natset: natset(n),
Set: Set{i:l},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
sq_type: SQType(T),
emptyset: {},
natset: natset(n),
or: P ∨ Q,
decidable: Dec(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
assert-bnot,
bool_cases_sqequal,
equal_wf,
lelt_wf,
decidable__lt,
or_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
plus-set_wf,
emptyset_wf,
primrec_wf,
setmem-plus-set,
not_wf,
bnot_wf,
assert_wf,
lt_int_wf,
primrec-unroll,
setmem-mkset-sq,
primrec0_lemma,
nat_wf,
primrec-wf2,
less_than_wf,
set_wf,
int_seg_subtype_nat,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
iff_wf,
all_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
Set_wf,
seteq_wf,
int_seg_wf,
exists_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
le_wf,
false_wf,
natset_wf,
set-subtype-coSet,
setmem_wf
Rules used in proof :
inrFormation,
inlFormation,
equalityElimination,
promote_hyp,
orFunctionality,
addLevel,
impliesFunctionality,
equalitySymmetry,
equalityTransitivity,
universeEquality,
cumulativity,
instantiate,
unionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
rename,
setElimination,
productElimination,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
independent_pairFormation,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x:Set\{i:l\}. ((x \mmember{} natset(n)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. seteq(x;natset(i)))
Date html generated:
2018_07_29-AM-10_03_03
Last ObjectModification:
2018_07_11-PM-05_49_21
Theory : constructive!set!theory
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