Nuprl Lemma : set-Leibniz-type
∀A:Type. (Leibniz-type{i:l}(A) ⇒ (∀B:A ⟶ ℙ. Leibniz-type{i:l}({a:A| B[a]} )))
Proof
Definitions occuring in Statement :
Leibniz-type: Leibniz-type{i:l}(T),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Leibniz-type: Leibniz-type{i:l}(T),
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
cand: A c∧ B,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
subtype_rel_dep_function,
istype-void,
Leibniz-type_wf,
istype-universe,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
cut,
hypothesisEquality,
applyEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality_alt,
functionEquality,
universeEquality,
universeIsType,
setEquality,
hypothesis,
because_Cache,
setIsType,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
independent_pairFormation,
independent_functionElimination,
voidElimination,
equalityIstype,
dependent_set_memberEquality_alt,
functionIsType,
productIsType,
unionIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination
Latex:
\mforall{}A:Type. (Leibniz-type\{i:l\}(A) {}\mRightarrow{} (\mforall{}B:A {}\mrightarrow{} \mBbbP{}. Leibniz-type\{i:l\}(\{a:A| B[a]\} )))
Date html generated:
2019_10_31-AM-07_26_08
Last ObjectModification:
2019_09_19-PM-06_48_11
Theory : constructive!algebra
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