Nuprl Lemma : no-excluded-middle-using-partial
¬(∀P:ℙ. (P ∨ (¬P)))
Proof
Definitions occuring in Statement :
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
false: False,
isl: isl(x),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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,
decidable: Dec(P),
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
has-value: (a)↓
Lemmas referenced :
istype-void,
int-value-type,
partial_wf,
fix_wf_partial,
int-mono,
eqtt_to_assert,
bottom_wf-partial,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
decidable__assert,
bottom_diverge,
sq_stable_from_decidable,
assert_wf,
has-value_wf-partial,
istype-assert,
int_subtype_base,
subtype_partial_sqtype_base,
sq_stable__has-value,
has-value_wf_base,
is-exception_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
functionIsType,
universeIsType,
universeEquality,
unionIsType,
sqequalHypSubstitution,
hypothesisEquality,
because_Cache,
cut,
introduction,
extract_by_obid,
hypothesis,
rename,
lambdaEquality_alt,
inhabitedIsType,
thin,
unionElimination,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isectElimination,
intEquality,
independent_isectElimination,
independent_pairFormation,
applyEquality,
setElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
natural_numberEquality,
equalityIstype,
imageMemberEquality,
baseClosed,
imageElimination,
divergentSqle,
sqleReflexivity
Latex:
\mneg{}(\mforall{}P:\mBbbP{}. (P \mvee{} (\mneg{}P)))
Date html generated:
2020_05_19-PM-09_36_45
Last ObjectModification:
2020_01_04-PM-07_57_41
Theory : partial_1
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