Nuprl Lemma : exists-pair
∀[A,B:Type]. ∀[P:(A × B) ⟶ ℙ]. (∃p:A × B. P[p] ⇐⇒ ∃a:A. ∃b:B. P[<a, b>])
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_pairEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_pairFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:(A \mtimes{} B) {}\mrightarrow{} \mBbbP{}]. (\mexists{}p:A \mtimes{} B. P[p] \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. \mexists{}b:B. P[<a, b>])
Date html generated:
2016_05_15-PM-03_47_16
Last ObjectModification:
2015_12_27-PM-01_21_17
Theory : general
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