Nuprl Lemma : fset-ac-le_transitivity
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac1,ac2,ac3:fset(fset(T))].
(fset-ac-le(eq;ac1;ac3)) supposing (fset-ac-le(eq;ac1;ac2) and fset-ac-le(eq;ac2;ac3))
Proof
Definitions occuring in Statement :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
fset-all: fset-all(s;x.P[x]),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
false: False,
all: ∀x:A. B[x],
top: Top,
guard: {T},
cand: A c∧ B,
squash: ↓T,
true: True
Lemmas referenced :
fset-all-iff,
fset_wf,
deq-fset_wf,
iff_weakening_uiff,
fset-all_wf,
bnot_wf,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
fset-member_wf,
assert_wf,
assert_witness,
deq_wf,
istype-universe,
assert_of_bnot,
equal-wf-T-base,
assert-fset-null,
istype-assert,
fset-extensionality,
mem_empty_lemma,
istype-void,
member-fset-filter,
assert-deq-f-subset,
fset-member_witness,
f-subset_transitivity,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :lambdaEquality_alt,
applyEquality,
setElimination,
rename,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
because_Cache,
isectEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
baseClosed,
voidElimination,
Error :equalityIstype,
sqequalBase,
dependent_functionElimination,
independent_pairFormation,
independent_pairEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[ac1,ac2,ac3:fset(fset(T))].
(fset-ac-le(eq;ac1;ac3)) supposing (fset-ac-le(eq;ac1;ac2) and fset-ac-le(eq;ac2;ac3))
Date html generated:
2019_06_20-PM-01_59_39
Last ObjectModification:
2019_01_01-AM-09_30_42
Theory : finite!sets
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