Nuprl Lemma : squash-exists-is-union-squash
∀[T:Type]. ∀[P:T ⟶ ℙ]. ↓∃x:T. P[x] ≡ ⋃x:T.(↓P[x])
Proof
Definitions occuring in Statement :
ext-eq: A ≡ B,
tunion: ⋃x:A.B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
tunion: ⋃x:A.B[x],
pi2: snd(t)
Lemmas referenced :
tunion_wf,
exists_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
applyEquality,
hypothesis,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
because_Cache,
imageElimination,
imageMemberEquality,
dependent_pairEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mdownarrow{}\mexists{}x:T. P[x] \mequiv{} \mcup{}x:T.(\mdownarrow{}P[x])
Date html generated:
2016_05_13-PM-04_13_59
Last ObjectModification:
2016_01_14-PM-07_29_10
Theory : subtype_1
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