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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