Nuprl Lemma : cosetTC-transitive
∀a:coSet{i:l}. transitive-set(cosetTC(a))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
cosetTC: cosetTC(a),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
set-item: set-item(s;x),
pi2: snd(t),
coW-item: coW-item(w;b),
true: True,
squash: ↓T,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
copath-extend: copath-extend(q;t),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
nat: ℕ,
coW-dom: coW-dom(a.B[a];w),
copath-at: copath-at(w;p),
pi1: fst(t),
set-dom: set-dom(s),
guard: {T},
subtype_rel: A ⊆r B,
coSet: coSet{i:l},
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cosetTC: cosetTC(a),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
iff_weakening_equal,
copath-at-extend,
true_wf,
squash_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
nat_wf,
copath-length_wf,
less_than_wf,
coW-dom_wf,
copath-extend_wf,
seteq_wf,
set-item_wf,
seteq_transitivity,
setmem-iff,
seteq_weakening,
setmem_functionality,
subtype_rel_self,
copath-at_wf,
setsubset_wf,
transitive-set-iff,
all_wf,
setsubset-iff,
cosetTC_wf,
coSet_wf,
setmem_wf,
setmem-mk-coset
Rules used in proof :
baseClosed,
imageMemberEquality,
equalityTransitivity,
imageElimination,
equalitySymmetry,
independent_pairFormation,
intEquality,
int_eqEquality,
approximateComputation,
independent_isectElimination,
unionElimination,
addEquality,
natural_numberEquality,
dependent_set_memberEquality,
dependent_pairFormation,
because_Cache,
applyEquality,
rename,
setElimination,
universeEquality,
cumulativity,
functionEquality,
lambdaEquality,
instantiate,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
allFunctionality,
addLevel,
hypothesisEquality,
sqequalRule,
productElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}a:coSet\{i:l\}. transitive-set(cosetTC(a))
Date html generated:
2018_07_29-AM-10_03_07
Last ObjectModification:
2018_07_18-PM-08_10_26
Theory : constructive!set!theory
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