Nuprl Lemma : set-axiom-of-choice-implies-xmiddle
Set-AC ⇒ (∀P:ℙ. ((↓P) ∨ (¬P)))
Proof
Definitions occuring in Statement :
set-axiom-of-choice: Set-AC,
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
set-axiom-of-choice: Set-AC,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
or: P ∨ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
top: Top,
eq_int: (i =z j),
ext-eq: A ≡ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
ifthenelse_wf,
eq_int_wf,
or_wf,
equal-wf-T-base,
set-axiom-of-choice_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
false_wf,
lelt_wf,
decidable__equal_int_seg,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
add-zero,
le-add-cancel,
not_wf,
squash_wf,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
lambdaEquality,
instantiate,
setElimination,
rename,
hypothesisEquality,
universeEquality,
setEquality,
intEquality,
baseClosed,
because_Cache,
independent_functionElimination,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
inlFormation,
applyEquality,
functionExtensionality,
applyLambdaEquality,
imageElimination,
addEquality,
isect_memberEquality,
voidEquality,
inrFormation
Latex:
Set-AC {}\mRightarrow{} (\mforall{}P:\mBbbP{}. ((\mdownarrow{}P) \mvee{} (\mneg{}P)))
Date html generated:
2017_04_14-AM-07_37_44
Last ObjectModification:
2017_02_27-PM-03_10_05
Theory : subtype_1
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