Nuprl Lemma : transitive-set_functionality
∀s1,s2:coSet{i:l}. (seteq(s1;s2) ⇒ (transitive-set(s1) ⇐⇒ transitive-set(s2)))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setsubset_functionality,
seteq_weakening,
setmem_functionality,
seteq_wf,
setsubset_wf,
setmem_wf,
coSet_wf,
all_wf,
iff_wf,
transitive-set_wf,
transitive-set-iff
Rules used in proof :
impliesLevelFunctionality,
allLevelFunctionality,
allFunctionality,
because_Cache,
functionEquality,
lambdaEquality,
sqequalRule,
instantiate,
isectElimination,
cumulativity,
independent_functionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
impliesFunctionality,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
addLevel,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s1,s2:coSet\{i:l\}. (seteq(s1;s2) {}\mRightarrow{} (transitive-set(s1) \mLeftarrow{}{}\mRightarrow{} transitive-set(s2)))
Date html generated:
2018_07_29-AM-10_02_49
Last ObjectModification:
2018_07_18-PM-01_32_02
Theory : constructive!set!theory
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