Nuprl Lemma : uni_sat_imp_in_uni_set
∀[T:Type]. ∀[a:T]. ∀[Q:T ⟶ ℙ]. ((a = !x:T. Q[x]) ⇒ (a ∈ {!x:T | Q[x]}))
Proof
Definitions occuring in Statement :
uni_sat: a = !x:T. Q[x],
unique_set: {!x:T | P[x]},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
unique_set: {!x:T | P[x]},
uni_sat: a = !x:T. Q[x]
Lemmas referenced :
all_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
hypothesisEquality,
independent_pairFormation,
hypothesis,
productEquality,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
universeEquality,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
because_Cache,
Error :functionIsType,
Error :inhabitedIsType,
Error :universeIsType,
cumulativity,
Error :isect_memberFormation_alt,
lambdaFormation,
productElimination,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[a:T]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. ((a = !x:T. Q[x]) {}\mRightarrow{} (a \mmember{} \{!x:T | Q[x]\}))
Date html generated:
2019_06_20-AM-11_18_13
Last ObjectModification:
2018_09_26-AM-10_25_20
Theory : core_2
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