Nuprl Lemma : exists-elim
∀[T:Type]. ∀[P:T ⟶ ℙ']. ∀a:T. ((∀x:T. (P[x] ⇒ (x = a ∈ T))) ⇒ {∃x:T. P[x] ⇐⇒ P[a]})
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
exists_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
addLevel,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
applyEquality,
levelHypothesis,
instantiate,
cumulativity,
lambdaEquality,
functionExtensionality,
dependent_pairFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}']. \mforall{}a:T. ((\mforall{}x:T. (P[x] {}\mRightarrow{} (x = a))) {}\mRightarrow{} \{\mexists{}x:T. P[x] \mLeftarrow{}{}\mRightarrow{} P[a]\})
Date html generated:
2017_10_01-AM-09_11_10
Last ObjectModification:
2017_07_26-PM-04_47_18
Theory : general
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