Nuprl Lemma : axiom-choice-0X-quot
∀B:{B:Type| B ⊆r Base} . ∀X:Type. ∀P:B ⟶ X ⟶ ℙ. ((∀n:B. ⇃(∃m:X. (P n m))) ⇒ ⇃(∃f:B ⟶ X. ∀n:B. (P n (f n))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
subtype_rel: A ⊆r B
Lemmas referenced :
subtype_rel_wf,
subtype_rel_set,
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
all_wf,
base_wf,
sq_stable__subtype_rel,
canonicalizable-base,
canonicalizable_wf,
trivial-quotient-true,
axiom-choice-quot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_functionElimination,
isectElimination,
hypothesis,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
cumulativity,
applyEquality,
because_Cache,
independent_isectElimination,
functionEquality,
instantiate,
universeEquality,
setEquality
Latex:
\mforall{}B:\{B:Type| B \msubseteq{}r Base\} . \mforall{}X:Type. \mforall{}P:B {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:B. \00D9(\mexists{}m:X. (P n m))) {}\mRightarrow{} \00D9(\mexists{}f:B {}\mrightarrow{} X. \mforall{}n:B. (P n (f n))))
Date html generated:
2016_05_14-PM-09_42_30
Last ObjectModification:
2016_01_15-PM-10_55_33
Theory : continuity
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