Nuprl Lemma : setmem-singleset
∀a,x:coSet{i:l}. ((x ∈ {a}) ⇐⇒ seteq(x;a))
Proof
Definitions occuring in Statement :
singleset: {a},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
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),
singleset: {a},
all: ∀x:A. B[x]
Lemmas referenced :
coSet_wf,
it_wf,
seteq_wf,
unit_wf2,
exists_wf,
setmem-mk-coset
Rules used in proof :
because_Cache,
dependent_pairFormation,
hypothesisEquality,
lambdaEquality,
productElimination,
independent_pairFormation,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,x:coSet\{i:l\}. ((x \mmember{} \{a\}) \mLeftarrow{}{}\mRightarrow{} seteq(x;a))
Date html generated:
2018_07_29-AM-09_53_17
Last ObjectModification:
2018_07_18-AM-10_55_36
Theory : constructive!set!theory
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