Nuprl Lemma : l_exists_single
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀x:T. ((∃y∈[x]. P[y]) ⇐⇒ P[x])
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
cons: [a / b],
nil: [],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
exists_wf,
member_singleton,
l_member_wf,
cons_wf,
nil_wf,
iff_wf,
l_exists_iff,
l_exists_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
hyp_replacement,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
applyEquality,
lambdaEquality,
productEquality,
because_Cache,
dependent_pairFormation,
addLevel,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
setEquality,
instantiate,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}x:T. ((\mexists{}y\mmember{}[x]. P[y]) \mLeftarrow{}{}\mRightarrow{} P[x])
Date html generated:
2019_06_20-PM-00_41_17
Last ObjectModification:
2018_08_24-PM-11_01_16
Theory : list_0
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