Nuprl Lemma : ax_choice
∀[A,B:Type]. ∀[Q:A ⟶ B ⟶ ℙ]. ((∀x:A. ∃y:B. Q[x;y]) ⇒ (∃f:A ⟶ B. ∀x:A. Q[x;f x]))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
pi1: fst(t)
Lemmas referenced :
all_wf,
exists_wf,
subtype_rel_self,
pi1_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
universeEquality,
Error :inhabitedIsType,
rename,
dependent_pairFormation,
productEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[Q:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:A. \mexists{}y:B. Q[x;y]) {}\mRightarrow{} (\mexists{}f:A {}\mrightarrow{} B. \mforall{}x:A. Q[x;f x]))
Date html generated:
2019_06_20-PM-00_26_27
Last ObjectModification:
2019_06_19-PM-06_41_26
Theory : fun_1
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