Nuprl Lemma : l_all-l_contains
∀[T:Type]. ∀[L1,L2:T List]. (L1 ⊆ L2 ⇒ (∀P:T ⟶ ℙ. ((∀x∈L2.P[x]) ⇒ (∀x∈L1.P[x]))))
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
l_all: (∀x∈L.P[x]),
l_contains: A ⊆ B,
member: t ∈ T,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
lelt_wf,
length_wf,
and_wf,
equal_wf,
int_seg_wf,
l_all_wf,
l_member_wf,
l_contains_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
rename,
setElimination,
dependent_set_memberEquality,
independent_pairFormation,
hypothesis,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
addLevel,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
applyEquality,
lambdaEquality,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
natural_numberEquality,
functionExtensionality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L1,L2:T List]. (L1 \msubseteq{} L2 {}\mRightarrow{} (\mforall{}P:T {}\mrightarrow{} \mBbbP{}. ((\mforall{}x\mmember{}L2.P[x]) {}\mRightarrow{} (\mforall{}x\mmember{}L1.P[x]))))
Date html generated:
2016_10_21-AM-10_05_25
Last ObjectModification:
2016_07_12-AM-05_25_17
Theory : list_1
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