Nuprl Lemma : isect2-b-union-subtype
∀[A,B,C:Type]. A ⋂ B ⋃ C ⊆r (A ⋂ B ⋃ A ⋂ C) supposing ¬B ⋂ C
Proof
Definitions occuring in Statement :
isect2: T1 ⋂ T2,
b-union: A ⋃ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
btrue: tt,
isect2: T1 ⋂ T2,
it: ⋅,
bfalse: ff,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
bfalse_wf,
strong-subtype-implies,
strong-subtype-b-union,
ifthenelse_wf,
bool_wf,
isect2_subtype_rel,
btrue_wf,
equal_wf,
isect2_subtype_rel2,
not_wf,
b-union_wf,
isect2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_pairFormation,
applyEquality,
productElimination,
imageElimination,
unionElimination,
equalityElimination,
imageMemberEquality,
dependent_pairEquality,
instantiate,
baseClosed,
rename,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[A,B,C:Type]. A \mcap{} B \mcup{} C \msubseteq{}r (A \mcap{} B \mcup{} A \mcap{} C) supposing \mneg{}B \mcap{} C
Date html generated:
2016_05_13-PM-04_12_04
Last ObjectModification:
2016_01_14-PM-07_29_55
Theory : subtype_1
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