Nuprl Lemma : setTC-unique
∀a,s:Set{i:l}.
((a ⊆ s) ⇒ transitive-set(s) ⇒ (∀s':Set{i:l}. ((a ⊆ s') ⇒ transitive-set(s') ⇒ (s ⊆ s'))) ⇒ seteq(s;setTC(a)))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setsubset: (a ⊆ b),
setTC: setTC(a),
seteq: seteq(s1;s2),
Set: Set{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
seteq-iff-setsubset,
setTC_wf,
setTC-contains,
setTC-transitive,
setTC-least,
all_wf,
Set_wf,
setsubset_wf,
transitive-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
instantiate,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality
Latex:
\mforall{}a,s:Set\{i:l\}.
((a \msubseteq{} s)
{}\mRightarrow{} transitive-set(s)
{}\mRightarrow{} (\mforall{}s':Set\{i:l\}. ((a \msubseteq{} s') {}\mRightarrow{} transitive-set(s') {}\mRightarrow{} (s \msubseteq{} s')))
{}\mRightarrow{} seteq(s;setTC(a)))
Date html generated:
2018_05_22-PM-09_51_39
Last ObjectModification:
2018_05_22-AM-11_25_11
Theory : constructive!set!theory
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