Nuprl Lemma : set-predicate_wf2
∀[s:Set{i:l}]. ∀[P:{x:Set{i:l}| (x ∈ s)} ⟶ ℙ']. (set-predicate{i:l}(s;x.P[x]) ∈ ℙ')
Proof
Definitions occuring in Statement :
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],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
exists: ∃x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
set-predicate: set-predicate{i:l}(s;a.P[a]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set-subtype-coSet,
Set_wf,
coSet-mem-Set-implies-Set,
seteq_wf,
setmem_wf,
coSet_wf,
all_wf
Rules used in proof :
isect_memberEquality,
universeEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_pairFormation,
independent_isectElimination,
dependent_set_memberEquality,
because_Cache,
applyEquality,
hypothesisEquality,
cumulativity,
functionEquality,
lambdaEquality,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[s:Set\{i:l\}]. \mforall{}[P:\{x:Set\{i:l\}| (x \mmember{} s)\} {}\mrightarrow{} \mBbbP{}']. (set-predicate\{i:l\}(s;x.P[x]) \mmember{} \mBbbP{}')
Date html generated:
2018_07_29-AM-09_52_10
Last ObjectModification:
2018_07_18-AM-10_07_49
Theory : constructive!set!theory
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