Nuprl Lemma : exists-product1
∀[A,B:Type]. ∀P:(A × B) ⟶ ℙ'. {∃x:A × B. P[x] ⇐⇒ ∃a:A. ∃b:B. P[<a, b>]}
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,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: 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 :
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
applyEquality,
independent_pairEquality,
cut,
instantiate,
lemma_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
productEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}P:(A \mtimes{} B) {}\mrightarrow{} \mBbbP{}'. \{\mexists{}x:A \mtimes{} B. P[x] \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. \mexists{}b:B. P[<a, b>]\}
Date html generated:
2016_05_15-PM-03_22_54
Last ObjectModification:
2015_12_27-PM-01_05_37
Theory : general
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