Nuprl Lemma : inl-inr-disjoint
∀[A,B:Type]. ∀[x:A]. ∀[y:B]. uiff((inl x) = (inr y ) ∈ (A + B);False)
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
false: False,
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
false: False,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
true: True,
prop: ℙ
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
equal_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
applyEquality,
lambdaEquality,
unionElimination,
thin,
natural_numberEquality,
unionEquality,
hypothesisEquality,
hypothesis,
instantiate,
lemma_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
promote_hyp,
because_Cache,
inlEquality,
inrEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[x:A]. \mforall{}[y:B]. uiff((inl x) = (inr y );False)
Date html generated:
2016_05_13-PM-03_20_15
Last ObjectModification:
2015_12_26-AM-09_10_58
Theory : union
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