Nuprl Lemma : axiom-choice-00-quot
∀P:ℕ ⟶ ℕ ⟶ ℙ. ((∀n:ℕ. ⇃(∃m:ℕ. (P n m))) ⇒ ⇃(∃f:ℕ ⟶ ℕ. ∀n:ℕ. (P n (f n))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
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,
nat: ℕ
Lemmas referenced :
axiom-choice-quot,
int_subtype_base,
le_wf,
set_subtype_base,
canonicalizable-base,
canonicalizable_wf,
trivial-quotient-true,
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
nat_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
hypothesisEquality,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
intEquality,
natural_numberEquality,
dependent_functionElimination
Latex:
\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}n:\mBbbN{}. \00D9(\mexists{}m:\mBbbN{}. (P n m))) {}\mRightarrow{} \00D9(\mexists{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n:\mBbbN{}. (P n (f n))))
Date html generated:
2016_05_14-PM-09_42_35
Last ObjectModification:
2016_01_06-PM-01_29_29
Theory : continuity
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