Nuprl Lemma : setmem-sub-coset
∀s:coSet{i:l}
∀[P:{a:coSet{i:l}| (a ∈ s)} ⟶ ℙ]
(set-predicate{i:l}(s;a.P[a]) ⇒ (∀a:coSet{i:l}. ((a ∈ {a ∈ s | P[a]}) ⇐⇒ (a ∈ s) ∧ P[a])))
Proof
Definitions occuring in Statement :
sub-set: {a ∈ s | P[a]},
set-predicate: set-predicate{i:l}(s;a.P[a]),
setmem: (x ∈ s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
sub-set: {a ∈ s | P[a]},
Wsup: Wsup(a;b),
mk-set: f"(T),
top: Top,
exists: ∃x:A. B[x],
pi1: fst(t),
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T},
set-predicate: set-predicate{i:l}(s;a.P[a]),
mk-coset: mk-coset(T;f),
set-item: set-item(s;x),
set-dom: set-dom(s),
pi2: snd(t)
Lemmas referenced :
subtype_coSet,
coSet_subtype,
setmem-mk-set-sq,
istype-void,
seteq_wf,
setmem_wf,
sub-set_wf,
subtype_rel_self,
set-predicate_wf,
coSet_wf,
seteq_inversion,
setmem-coset,
setmem-iff,
mk-coset_wf,
seteq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
independent_pairFormation,
hypothesis_subsumption,
cut,
introduction,
extract_by_obid,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
isectElimination,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
universeIsType,
lambdaEquality_alt,
cumulativity,
inhabitedIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
setIsType,
productIsType,
instantiate,
universeEquality,
because_Cache,
functionIsType,
dependent_functionElimination,
independent_functionElimination,
dependent_pairEquality_alt
Latex:
\mforall{}s:coSet\{i:l\}
\mforall{}[P:\{a:coSet\{i:l\}| (a \mmember{} s)\} {}\mrightarrow{} \mBbbP{}]
(set-predicate\{i:l\}(s;a.P[a]) {}\mRightarrow{} (\mforall{}a:coSet\{i:l\}. ((a \mmember{} \{a \mmember{} s | P[a]\}) \mLeftarrow{}{}\mRightarrow{} (a \mmember{} s) \mwedge{} P[a])))
Date html generated:
2019_10_31-AM-06_33_16
Last ObjectModification:
2018_11_12-AM-09_30_54
Theory : constructive!set!theory
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