Nuprl Lemma : all-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],
iff: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
all_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
hypothesisEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
unionElimination,
inlEquality,
inrEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}P:(A + B) {}\mrightarrow{} \mBbbP{}. (\mforall{}x:A + B. P[x] \mLeftarrow{}{}\mRightarrow{} (\mforall{}a:A. P[inl a]) \mwedge{} (\mforall{}b:B. P[inr b ]))
Date html generated:
2016_05_15-PM-03_24_39
Last ObjectModification:
2015_12_27-PM-01_06_08
Theory : general
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