Nuprl Lemma : prank-psub
∀a,b:formula(). (a ⊆ b ⇒ ((a = b ∈ formula()) ∨ prank(a) < prank(b)))
Proof
Definitions occuring in Statement :
psub: a ⊆ b,
prank: prank(x),
formula: formula(),
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
prank: prank(x),
psub: a ⊆ b,
pvar: pvar(name),
formula_ind: formula_ind,
or: P ∨ Q,
pnot: pnot(sub),
guard: {T},
pand: pand(left;right),
por: por(left;right),
pimp: pimp(left;right),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
formula_wf,
formula-induction,
psub_wf,
or_wf,
equal_wf,
less_than_wf,
prank_wf,
nat_wf,
equal-wf-T-base,
atom_subtype_base,
pnot_wf,
imax_wf,
pand_wf,
por_wf,
pimp_wf,
ifthenelse_wf,
le_int_wf,
bool_wf,
uiff_transitivity,
assert_wf,
le_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lt_int_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imax_unfold,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
inlFormation,
natural_numberEquality,
baseApply,
closedConclusion,
baseClosed,
atomEquality,
unionElimination,
addEquality,
inrFormation,
intEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
hyp_replacement,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}a,b:formula(). (a \msubseteq{} b {}\mRightarrow{} ((a = b) \mvee{} prank(a) < prank(b)))
Date html generated:
2018_05_21-PM-08_53_06
Last ObjectModification:
2017_07_26-PM-06_16_34
Theory : general
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