Nuprl Lemma : f-proper-subset_transitivity
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs,cs:fset(T). (as ⊆≠ bs ⇒ bs ⊆≠ cs ⇒ as ⊆≠ cs)
Proof
Definitions occuring in Statement :
f-proper-subset: xs ⊆≠ ys,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
f-proper-subset: xs ⊆≠ ys,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
f-subset: xs ⊆ ys,
uimplies: b supposing a,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
fset-size-proper-subset,
f-proper-subset_wf,
fset_wf,
deq_wf,
fset-member_witness,
fset-member_wf,
equal_wf,
f-subset_transitivity,
less_than_wf,
fset-size_wf,
nat_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
independent_pairFormation,
cumulativity,
sqequalRule,
lambdaEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
universeEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
setElimination,
rename,
imageElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs,cs:fset(T). (as \msubseteq{}\mneq{} bs {}\mRightarrow{} bs \msubseteq{}\mneq{} cs {}\mRightarrow{} as \msubseteq{}\mneq{} cs)
Date html generated:
2017_04_17-AM-09_22_30
Last ObjectModification:
2017_02_27-PM-05_24_24
Theory : finite!sets
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