Nuprl Lemma : setTC-transitive
∀a:Set{i:l}. transitive-set(setTC(a))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setTC: setTC(a),
Set: Set{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
top: Top,
cand: A c∧ B,
exists: ∃x:A. B[x],
uimplies: b supposing a,
or: P ∨ Q,
guard: {T},
setTC: Error :setTC,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
setmem-mk-set,
setTC_functionality,
seteq_weakening,
setmem_functionality,
coSet-seteq-Set,
setmem-mk-set-sq,
setTC-contains,
setTC-set-function,
setmem-setunionfun,
set-add_wf,
setmem-set-add,
coSet-mem-Set-implies-Set,
setunionfun_wf,
or_wf,
mk-set_wf,
setsubset_wf,
transitive-set-iff,
setsubset-iff,
set-subtype-coSet,
Set_wf,
setmem_wf,
coSet_wf,
all_wf,
set-induction
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
productEquality,
independent_pairFormation,
unionElimination,
setEquality,
dependent_pairFormation,
independent_isectElimination,
rename,
setElimination,
functionExtensionality,
inrFormation,
productElimination,
dependent_functionElimination,
impliesFunctionality,
allFunctionality,
addLevel,
universeEquality,
lambdaFormation,
independent_functionElimination,
because_Cache,
applyEquality,
hypothesisEquality,
cumulativity,
functionEquality,
hypothesis,
instantiate,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}a:Set\{i:l\}. transitive-set(setTC(a))
Date html generated:
2018_07_29-AM-10_03_39
Last ObjectModification:
2018_07_18-PM-05_02_09
Theory : constructive!set!theory
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