Nuprl Lemma : natset-seteq-natset
∀n,m:ℕ. (seteq(natset(n);natset(m)) ⇐⇒ n = m ∈ ℤ)
Proof
Definitions occuring in Statement :
natset: natset(n),
seteq: seteq(s1;s2),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
label: ...$L... t,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
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 :
iff_weakening_equal,
subtype_rel_self,
Set_wf,
true_wf,
squash_wf,
seteq_weakening,
le_wf,
int_term_value_constant_lemma,
itermConstant_wf,
decidable__le,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
natset-subset-natset,
seteq-iff-setsubset,
nat_wf,
equal_wf,
natset_wf,
seteq_wf
Rules used in proof :
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
dependent_set_memberEquality,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
promote_hyp,
because_Cache,
independent_functionElimination,
productElimination,
dependent_functionElimination,
rename,
setElimination,
intEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n,m:\mBbbN{}. (seteq(natset(n);natset(m)) \mLeftarrow{}{}\mRightarrow{} n = m)
Date html generated:
2018_05_29-PM-01_49_49
Last ObjectModification:
2018_05_25-AM-00_06_58
Theory : constructive!set!theory
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