Nuprl Lemma : cosetTC-least
∀a,s:coSet{i:l}. ((a ⊆ s) ⇒ transitive-set(s) ⇒ (cosetTC(a) ⊆ s))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setsubset: (a ⊆ b),
cosetTC: cosetTC(a),
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
allsetmem: ∀a∈A.P[a],
setsubset: (a ⊆ b),
set-dom: set-dom(s),
coW-dom: coW-dom(a.B[a];w),
eq_int: (i =z j),
subtract: n - m,
set-item: set-item(s;x),
pi2: snd(t),
coW-item: coW-item(w;b),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
coPath-at: coPath-at(n;w;p),
coPath: coPath(a.B[a];w;n),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
false: False,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
copath-at: copath-at(w;p),
pi1: fst(t),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
prop: ℙ,
subtype_rel: A ⊆r B,
coSet: coSet{i:l},
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
top: Top,
cosetTC: cosetTC(a),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
transitive-set-iff,
set-item_wf,
decidable__lt,
set-dom_wf,
int_subtype_base,
decidable__equal_int,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
top_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
nat_wf,
primrec-wf2,
set_wf,
coPath-at_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
all_wf,
subtract_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
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,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
coPath_wf,
less_than_wf,
coSet_wf,
setsubset_wf,
transitive-set_wf,
setmem_wf,
subtype_rel_self,
copath-at_wf,
setmem_functionality_1,
setmem-mk-coset,
setsubset-iff,
cosetTC_functionality_subset,
cosetTC_wf,
setsubset_transitivity
Rules used in proof :
promote_hyp,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
functionEquality,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_set_memberEquality,
cumulativity,
natural_numberEquality,
imageElimination,
instantiate,
applyEquality,
rename,
setElimination,
lambdaEquality,
universeEquality,
because_Cache,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
productElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,s:coSet\{i:l\}. ((a \msubseteq{} s) {}\mRightarrow{} transitive-set(s) {}\mRightarrow{} (cosetTC(a) \msubseteq{} s))
Date html generated:
2018_07_29-AM-10_03_11
Last ObjectModification:
2018_07_18-PM-08_43_25
Theory : constructive!set!theory
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