Nuprl Lemma : setmem-sub-set
∀s:Set{i:l}
∀[P:{a:Set{i:l}| (a ∈ s)} ⟶ ℙ]
(set-predicate{i:l}(s;a.P[a]) ⇒ (∀a:Set{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]),
Set: Set{i:l},
setmem: (x ∈ s),
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 :
set-predicate: set-predicate{i:l}(s;a.P[a]),
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
pi1: fst(t),
exists: ∃x:A. B[x],
top: Top,
sub-set: {a ∈ s | P[a]},
Wsup: Wsup(a;b),
mk-set: f"(T),
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
seteq_wf,
item_mk_set_lemma,
dom_mk_set_lemma,
setmem-iff,
set-predicate-iff,
seteq_inversion,
set-predicate_wf2,
Set_wf,
subtype_rel_self,
coSet_wf,
coSet-mem-Set-implies-Set,
sub-set_wf,
setmem_wf,
setmem-mk-set,
set-subtype-coSet,
mk-set_wf,
setmem_functionality_1,
setmem-mk-set-sq,
set-subtype,
subtype-set
Rules used in proof :
functionExtensionality,
dependent_pairEquality,
functionEquality,
universeEquality,
instantiate,
productEquality,
cumulativity,
setEquality,
dependent_pairFormation,
independent_isectElimination,
dependent_set_memberEquality,
setElimination,
lambdaEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
isectElimination,
rename,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
independent_pairFormation,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s:Set\{i:l\}
\mforall{}[P:\{a:Set\{i:l\}| (a \mmember{} s)\} {}\mrightarrow{} \mBbbP{}]
(set-predicate\{i:l\}(s;a.P[a]) {}\mRightarrow{} (\mforall{}a:Set\{i:l\}. ((a \mmember{} \{a \mmember{} s | P[a]\}) \mLeftarrow{}{}\mRightarrow{} (a \mmember{} s) \mwedge{} P[a])))
Date html generated:
2018_07_29-AM-09_52_25
Last ObjectModification:
2018_07_18-AM-10_24_57
Theory : constructive!set!theory
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