Nuprl Lemma : uall-union
∀[A,B:Type]. ∀P:(A + B) ⟶ ℙ. ((∀[x:A + B]. P[x]) ⇒ ((∀[a:A]. P[inl a]) ∧ (∀[b:B]. P[inr b ])))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
independent_pairFormation,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
inlEquality,
inrEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}P:(A + B) {}\mrightarrow{} \mBbbP{}. ((\mforall{}[x:A + B]. P[x]) {}\mRightarrow{} ((\mforall{}[a:A]. P[inl a]) \mwedge{} (\mforall{}[b:B]. P[inr b ])))
Date html generated:
2016_05_15-PM-03_24_45
Last ObjectModification:
2015_12_27-PM-01_06_11
Theory : general
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