Nuprl Lemma : false-sqequal
∀[a,b:Base]. ((¬(a ~ b)) ⇒ ((a ~ b) = (0 ~ 1) ∈ Type))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
base: Base,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
subtype_base_sq,
base_wf,
subtype_rel_self,
false_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalExtensionalEquality,
independent_pairFormation,
sqequalHypSubstitution,
independent_functionElimination,
thin,
hypothesis,
voidElimination,
Error :universeIsType,
because_Cache,
instantiate,
extract_by_obid,
isectElimination,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
hypothesisEquality,
sqequalIntensionalEquality,
baseClosed,
sqequalRule,
imageMemberEquality,
Error :functionIsType,
Error :lambdaEquality_alt,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
intEquality,
natural_numberEquality
Latex:
\mforall{}[a,b:Base]. ((\mneg{}(a \msim{} b)) {}\mRightarrow{} ((a \msim{} b) = (0 \msim{} 1)))
Date html generated:
2019_06_20-AM-11_19_43
Last ObjectModification:
2018_10_15-PM-10_52_07
Theory : sqequal_1
Home
Index