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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