Nuprl Lemma : decidable__equal_set
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀[P:T ⟶ Type]. ∀x,y:{x:T| P[x]} . Dec(x = y ∈ {x:T| P[x]} )))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
not: ¬A,
subtype_rel: A ⊆r B,
label: ...$L... t,
false: False
Lemmas referenced :
it_wf,
not_wf,
equal_wf,
decidable_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
setEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
unionElimination,
inlFormation,
inrFormation,
dependent_set_memberEquality,
introduction,
equalityElimination,
independent_functionElimination,
because_Cache,
voidElimination
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}[P:T {}\mrightarrow{} Type]. \mforall{}x,y:\{x:T| P[x]\} . Dec(x = y)))
Date html generated:
2016_05_13-PM-03_17_57
Last ObjectModification:
2016_01_06-PM-05_20_29
Theory : core_2
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