Nuprl Lemma : setmem-pairset
∀a,b,x:coSet{i:l}. ((x ∈ {a,b}) ⇐⇒ seteq(x;a) ∨ seteq(x;b))
Proof
Definitions occuring in Statement :
pairset: {a,b},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
bfalse: ff,
btrue: tt,
guard: {T},
or: P ∨ Q,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x],
mk-coset: mk-coset(T;f),
pairset: {a,b},
all: ∀x:A. B[x]
Lemmas referenced :
bfalse_wf,
btrue_wf,
or_wf,
coSet_wf,
ifthenelse_wf,
seteq_wf,
bool_wf,
exists_wf,
setmem-mk-coset
Rules used in proof :
dependent_pairFormation,
inrFormation,
inlFormation,
unionElimination,
productElimination,
because_Cache,
instantiate,
applyEquality,
hypothesisEquality,
lambdaEquality,
independent_pairFormation,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b,x:coSet\{i:l\}. ((x \mmember{} \{a,b\}) \mLeftarrow{}{}\mRightarrow{} seteq(x;a) \mvee{} seteq(x;b))
Date html generated:
2018_07_29-AM-09_59_35
Last ObjectModification:
2018_07_18-AM-11_09_00
Theory : constructive!set!theory
Home
Index