Nuprl Lemma : b-union-equality-disjoint
∀A,B:Type. ∀a:A. ∀b:B. ((¬A ⋂ B) ⇒ (¬(a = b ∈ (A ⋃ B))))
Proof
Definitions occuring in Statement :
isect2: T1 ⋂ T2,
b-union: A ⋃ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
isect2: T1 ⋂ T2,
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
tunion: ⋃x:A.B[x],
b-union: A ⋃ B,
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
guard: {T},
uimplies: b supposing a,
pi1: fst(t),
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q
Lemmas referenced :
istype-universe,
istype-void,
isect2_wf,
subtype_rel_b-union-right,
subtype_rel_b-union-left,
b-union_wf,
bool_wf,
bfalse_wf,
ifthenelse_wf,
pi2_wf,
top_wf,
istype-top,
pair-eta,
subtype_rel_product,
pi1_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
universeEquality,
instantiate,
Error :inhabitedIsType,
Error :functionIsType,
applyEquality,
isectElimination,
Error :universeIsType,
Error :equalityIstype,
because_Cache,
voidElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
sqequalRule,
equalityElimination,
unionElimination,
isect_memberEquality,
introduction,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
thin,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageEqInduction,
baseClosed,
Error :dependent_pairEquality_alt,
Error :lambdaEquality_alt,
Error :productIsType,
independent_pairEquality,
Error :isect_memberEquality_alt,
independent_isectElimination,
applyLambdaEquality,
dependent_functionElimination,
cumulativity,
productElimination
Latex:
\mforall{}A,B:Type. \mforall{}a:A. \mforall{}b:B. ((\mneg{}A \mcap{} B) {}\mRightarrow{} (\mneg{}(a = b)))
Date html generated:
2019_06_20-PM-00_28_01
Last ObjectModification:
2019_01_02-PM-03_16_14
Theory : subtype_1
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