Nuprl Lemma : no_repeats-settype
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[L:{x:T| P[x]}  List].  uiff(no_repeats(T;L);no_repeats({x:T| P[x]} L))
Proof
Definitions occuring in Statement : 
no_repeats: no_repeats(T;l)
, 
list: T List
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
implies: P 
⇒ Q
, 
so_lambda: λ2x.t[x]
Lemmas referenced : 
no_repeats-subtype, 
no_repeats_witness, 
no_repeats_wf, 
subtype_rel_list, 
no_repeats-strong-subtype, 
strong-subtype-set3, 
strong-subtype-self, 
list_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
independent_pairFormation, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
setEquality, 
applyEquality, 
hypothesis, 
lambdaEquality, 
sqequalRule, 
universeEquality, 
independent_isectElimination, 
setElimination, 
rename, 
because_Cache, 
independent_functionElimination, 
productElimination, 
independent_pairEquality, 
isect_memberEquality, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
cumulativity
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[L:\{x:T|  P[x]\}    List].    uiff(no\_repeats(T;L);no\_repeats(\{x:T|  P[x]\}  ;L))
Date html generated:
2016_05_14-PM-01_27_06
Last ObjectModification:
2015_12_26-PM-04_50_39
Theory : list_1
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