Nuprl Lemma : natset-subset-natset
∀n,m:ℕ. ((natset(n) ⊆ natset(m)) ⇐⇒ n ≤ m)
Proof
Definitions occuring in Statement :
natset: natset(n),
setsubset: (a ⊆ b),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
Set_wf,
setmem_wf,
int_term_value_constant_lemma,
itermConstant_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
transitive-set-iff,
natset-transitive,
setmem-irreflexive,
setsubset-iff,
natset-setmem-natset,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
decidable__lt,
nat_wf,
le_wf,
natset_wf,
setsubset_wf
Rules used in proof :
because_Cache,
dependent_set_memberEquality,
cumulativity,
instantiate,
productElimination,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n,m:\mBbbN{}. ((natset(n) \msubseteq{} natset(m)) \mLeftarrow{}{}\mRightarrow{} n \mleq{} m)
Date html generated:
2018_05_29-PM-01_49_46
Last ObjectModification:
2018_05_25-AM-00_06_25
Theory : constructive!set!theory
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