Nuprl Lemma : product-union-atom-disjoint
∀[T,S,A,B:Type]. (¬T × S ⋂ (A + B) ⋃ Atom)
Proof
Definitions occuring in Statement :
isect2: T1 ⋂ T2,
b-union: A ⋃ B,
uall: ∀[x:A]. B[x],
not: ¬A,
product: x:A × B[x],
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
all: ∀x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
pi2: snd(t),
bfalse: ff,
prop: ℙ,
has-value: (a)↓
Lemmas referenced :
isect2_decomp,
b-union_wf,
pair-eta,
isect2_subtype_rel3,
top_wf,
subtype_rel_product,
subtype_rel_wf,
isatom-implies-not-ispair,
atom_subtype_base,
value-type-has-value,
atom-value-type,
assert_wf,
bfalse_wf,
has-value_wf_base,
is-exception_wf,
btrue_wf,
equal_wf,
isect2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
cumulativity,
hypothesisEquality,
unionEquality,
atomEquality,
hypothesis,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
sqequalRule,
applyEquality,
because_Cache,
independent_isectElimination,
inlFormation,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageElimination,
unionElimination,
equalityElimination,
isatomReduceTrue,
independent_functionElimination,
ispairCases,
divergentSqle,
instantiate,
baseClosed,
sqequalAxiom,
dependent_functionElimination,
universeEquality
Latex:
\mforall{}[T,S,A,B:Type]. (\mneg{}T \mtimes{} S \mcap{} (A + B) \mcup{} Atom)
Date html generated:
2018_05_21-PM-10_19_11
Last ObjectModification:
2017_07_26-PM-06_36_52
Theory : eval!all
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