Nuprl Lemma : member-p-union
∀p:FinProbSpace. ∀A,B:p-open(p). ∀s:ℕ ⟶ Outcome.  (s ∈ p-union(A;B) 
⇐⇒ s ∈ A ∨ s ∈ B)
Proof
Definitions occuring in Statement : 
p-union: p-union(A;B)
, 
p-open-member: s ∈ C
, 
p-open: p-open(p)
, 
p-outcome: Outcome
, 
finite-prob-space: FinProbSpace
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
p-open-member: s ∈ C
, 
p-union: p-union(A;B)
, 
p-open: p-open(p)
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
so_lambda: λ2x.t[x]
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s]
, 
nat: ℕ
, 
uimplies: b supposing a
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
int_seg: {i..j-}
, 
rev_implies: P 
⇐ Q
, 
eq_int: (i =z j)
Lemmas referenced : 
exists_wf, 
nat_wf, 
equal-wf-T-base, 
eq_int_wf, 
subtype_rel_dep_function, 
p-outcome_wf, 
int_seg_wf, 
int_seg_subtype_nat, 
false_wf, 
subtype_rel_self, 
bool_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
or_wf, 
set_wf, 
all_wf, 
le_wf, 
finite-prob-space_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
bfalse_wf, 
assert_elim, 
and_wf, 
btrue_neq_bfalse, 
bool_cases, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot, 
int_subtype_base, 
uiff_transitivity
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
lambdaFormation, 
independent_pairFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesis, 
lambdaEquality, 
because_Cache, 
applyEquality, 
setElimination, 
rename, 
hypothesisEquality, 
dependent_pairEquality, 
natural_numberEquality, 
independent_isectElimination, 
functionEquality, 
unionElimination, 
equalityElimination, 
productElimination, 
equalityTransitivity, 
equalitySymmetry, 
dependent_pairFormation, 
promote_hyp, 
dependent_functionElimination, 
instantiate, 
cumulativity, 
independent_functionElimination, 
voidElimination, 
baseClosed, 
productEquality, 
functionExtensionality, 
addLevel, 
levelHypothesis, 
dependent_set_memberEquality, 
applyLambdaEquality, 
impliesFunctionality, 
inlFormation, 
inrFormation, 
intEquality
Latex:
\mforall{}p:FinProbSpace.  \mforall{}A,B:p-open(p).  \mforall{}s:\mBbbN{}  {}\mrightarrow{}  Outcome.    (s  \mmember{}  p-union(A;B)  \mLeftarrow{}{}\mRightarrow{}  s  \mmember{}  A  \mvee{}  s  \mmember{}  B)
Date html generated:
2018_05_22-AM-00_36_37
Last ObjectModification:
2017_07_26-PM-07_00_31
Theory : randomness
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