Nuprl Lemma : setmem-set-add
∀a,b,x:coSet{i:l}. ((x ∈ a + b) ⇐⇒ (x ∈ a) ∨ (x ∈ b))
Proof
Definitions occuring in Statement :
set-add: a + b,
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
set-add: a + b,
mk-coset: mk-coset(T;f),
uall: ∀[x:A]. B[x],
top: Top,
exists: ∃x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
subtype_coSet,
coSet_subtype,
setmem-mk-coset,
setmem_wf,
set-add_wf,
mk-coset_wf,
or_wf,
coSet_wf,
seteq_wf,
exists_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis_subsumption,
cut,
introduction,
extract_by_obid,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
independent_pairFormation,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
unionElimination,
inlFormation,
dependent_pairFormation,
lambdaEquality,
inrFormation,
inlEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
dependent_functionElimination,
independent_functionElimination,
inrEquality
Latex:
\mforall{}a,b,x:coSet\{i:l\}. ((x \mmember{} a + b) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} a) \mvee{} (x \mmember{} b))
Date html generated:
2019_10_31-AM-06_33_24
Last ObjectModification:
2018_08_21-PM-02_01_22
Theory : constructive!set!theory
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