Nuprl Lemma : list-equal-set2
∀[T:Type]. ∀[P:T ⟶ ℙ].
∀[L:{x:T| P[x]} List]. ∀[L':T List]. L = L' ∈ ({x:T| P[x]} List) supposing L = L' ∈ (T List)
supposing ∀x:T. SqStable(P[x])
Proof
Definitions occuring in Statement :
list: T List,
sq_stable: SqStable(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
strong-subtype-equal-lists,
strong-subtype-set3,
strong-subtype-self,
equal_wf,
list_wf,
subtype_rel_list,
all_wf,
sq_stable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
sqequalRule,
universeEquality,
independent_isectElimination,
because_Cache,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
\mforall{}[L:\{x:T| P[x]\} List]. \mforall{}[L':T List]. L = L' supposing L = L' supposing \mforall{}x:T. SqStable(P[x])
Date html generated:
2016_05_14-AM-07_49_14
Last ObjectModification:
2015_12_26-PM-04_45_02
Theory : list_1
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