Nuprl Lemma : assert-f-proper-subset-dec
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[xs,ys:fset(T)]. uiff(↑f-proper-subset-dec(eq;xs;ys);xs ⊆≠ ys)
Proof
Definitions occuring in Statement :
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
f-proper-subset: xs ⊆≠ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
f-proper-subset: xs ⊆≠ ys,
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
f-subset: xs ⊆ ys,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
deq: EqDecider(T)
Lemmas referenced :
equal_wf,
fset_wf,
fset-member_witness,
fset-member_wf,
f-subset_wf,
not_wf,
assert_wf,
band_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
bnot_wf,
deq-fset_wf,
deq_wf,
uiff_wf,
f-proper-subset-dec_wf,
f-proper-subset_wf,
assert_witness,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert-deq-f-subset,
assert_of_bnot,
assert-deq-fset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lambdaFormation,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
because_Cache,
applyEquality,
setElimination,
rename,
setEquality,
functionEquality,
functionExtensionality,
universeEquality,
addLevel,
independent_isectElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[xs,ys:fset(T)]. uiff(\muparrow{}f-proper-subset-dec(eq;xs;ys);xs \msubseteq{}\mneq{} ys)
Date html generated:
2017_04_17-AM-09_20_23
Last ObjectModification:
2017_02_27-PM-05_23_18
Theory : finite!sets
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