Nuprl Lemma : setmem-emptyset
∀x:coSet{i:l}. ((x ∈ {}) ⇐⇒ False)
Proof
Definitions occuring in Statement :
emptyset: {},
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
prop: ℙ,
exists: ∃x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
so_lambda: λ2x.t[x],
member: t ∈ T,
emptyset: {},
false: False,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
coSet_wf,
false_wf,
emptyset_wf,
set-subtype-coSet,
setmem_wf,
it_wf,
mkset_wf,
coSet-mem-Set-implies-Set,
setmem-mkset
Rules used in proof :
independent_functionElimination,
productElimination,
lambdaEquality,
because_Cache,
dependent_pairFormation,
independent_isectElimination,
hypothesisEquality,
isectElimination,
hypothesis,
isect_memberEquality,
voidElimination,
applyEquality,
functionExtensionality,
sqequalRule,
voidEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
sqequalHypSubstitution,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x:coSet\{i:l\}. ((x \mmember{} \{\}) \mLeftarrow{}{}\mRightarrow{} False)
Date html generated:
2018_07_29-AM-09_53_07
Last ObjectModification:
2018_07_18-AM-10_42_59
Theory : constructive!set!theory
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