Nuprl Lemma : transitive-set-iff2
∀s:Set{i:l}. (transitive-set(s) ⇐⇒ ∀x:Set{i:l}. ((x ∈ s) ⇒ (x ⊆ s)))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setsubset: (a ⊆ b),
Set: Set{i:l},
setmem: (x ∈ s),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
set-predicate: set-predicate{i:l}(s;a.P[a]),
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
transitive-set: transitive-set(s),
all: ∀x:A. B[x]
Lemmas referenced :
coSet-mem-Set-implies-Set,
iff_wf,
allsetmem_wf,
seteq_wf,
seteq_weakening,
setsubset_functionality,
allsetmem-iff,
setsubset_wf,
coSet_wf,
all_wf,
Set_wf,
set-subtype-coSet,
setmem_wf
Rules used in proof :
dependent_pairFormation,
independent_isectElimination,
independent_functionElimination,
setEquality,
rename,
setElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
addLevel,
functionEquality,
cumulativity,
lambdaEquality,
instantiate,
because_Cache,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s:Set\{i:l\}. (transitive-set(s) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:Set\{i:l\}. ((x \mmember{} s) {}\mRightarrow{} (x \msubseteq{} s)))
Date html generated:
2018_07_29-AM-10_02_46
Last ObjectModification:
2018_07_18-PM-10_06_18
Theory : constructive!set!theory
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