Nuprl Lemma : setTC_wf
∀[a:Set{i:l}]. (setTC(a) ∈ Set{i:l})
Proof
Definitions occuring in Statement :
setTC: setTC(a),
Set: Set{i:l},
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
Set: Set{i:l},
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step,
spreadn: spread3,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
ext-eq: A ≡ B,
unit: Unit,
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
pi1: fst(t),
nat_plus: ℕ+,
W-rel: W-rel(A;a.B[a];w),
param-W-rel: param-W-rel,
pcw-steprel: StepRel(s1;s2),
pi2: snd(t),
isl: isl(x),
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B,
guard: {T},
Wsup: Wsup(a;b),
sq_type: SQType(T),
le: A ≤ B,
sq_stable: SqStable(P),
mk-set: f"(T),
setTC: setTC(a),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
Set_wf,
W-elimination-facts,
istype-universe,
subtype_rel_self,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
istype-top,
istype-void,
istype-true,
add-subtract-cancel,
itermAdd_wf,
int_term_value_add_lemma,
W-ext,
param-co-W-ext,
unit_wf2,
it_wf,
param-co-W_wf,
top_wf,
pcw-steprel_wf,
true_wf,
false_wf,
subtype_rel_dep_function,
less_than_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_function,
int_seg_wf,
int_seg_subtype,
istype-false,
sq_stable__le,
setmem_wf,
set-subtype-coSet,
mk-set_wf,
setmem-mk-set-sq,
set-add_wf2,
setunionfun_wf2,
setmem_functionality,
seteq_weakening,
seteq_inversion,
plus-set_wf2,
plus-set_wf,
setmem-plus-set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
thin,
lambdaFormation_alt,
hypothesisEquality,
instantiate,
dependent_functionElimination,
universeEquality,
sqequalRule,
lambdaEquality_alt,
cumulativity,
isectElimination,
productElimination,
strong_bar_Induction,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_functionElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
productIsType,
because_Cache,
inhabitedIsType,
lessCases,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
closedConclusion,
axiomEquality,
equalityIstype,
addEquality,
int_eqReduceTrueSq,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
dependent_pairEquality_alt,
inlEquality_alt,
unionIsType,
productEquality,
unionEquality,
hyp_replacement,
applyLambdaEquality,
intEquality,
setIsType,
inrFormation,
dependent_set_memberEquality
Latex:
\mforall{}[a:Set\{i:l\}]. (setTC(a) \mmember{} Set\{i:l\})
Date html generated:
2020_05_20-PM-01_18_42
Last ObjectModification:
2020_01_06-PM-01_24_39
Theory : constructive!set!theory
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