Nuprl Lemma : FormSet_wf2
∀[C:Type]. ∀[x:Atom]. ∀[phi:PZF-Formula(C)]. {x | phi} ∈ PZF-Term(C) supposing PZF-safe(phi;[x])
Proof
Definitions occuring in Statement :
PZF-Formula: PZF-Formula(C),
PZF-Term: PZF-Term(C),
PZF-safe: PZF-safe(phi;vs),
FormSet: {var | phi},
cons: [a / b],
nil: [],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
PZF-Formula: PZF-Formula(C),
PZF-Form: PZF-Form(C),
PZF-Term: PZF-Term(C),
and: P ∧ Q,
cand: A c∧ B,
wfForm: wfForm(f),
termForm: termForm(f),
wfFormAux: wfFormAux(f),
FormSet: {var | phi},
Form_ind: Form_ind,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
btrue: tt,
SafeForm: SafeForm(f),
prop: ℙ,
assert: ↑b,
true: True,
squash: ↓T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
FormSet_wf,
assert_wf,
wfForm_wf,
SafeForm_wf,
termForm_wf,
PZF-safe_wf,
cons_wf,
nil_wf,
PZF-Formula_wf,
squash_wf,
true_wf,
bool_wf,
wfFormAux_wf,
eqtt_to_assert,
not_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
false_wf,
assert_of_band,
PZF_safe_wf,
assert-PZF_safe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
independent_pairFormation,
productEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
atomEquality,
isect_memberEquality,
because_Cache,
universeEquality,
hyp_replacement,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate
Latex:
\mforall{}[C:Type]. \mforall{}[x:Atom]. \mforall{}[phi:PZF-Formula(C)]. \{x | phi\} \mmember{} PZF-Term(C) supposing PZF-safe(phi;[x])
Date html generated:
2018_05_21-PM-11_37_39
Last ObjectModification:
2017_10_12-PM-04_50_14
Theory : PZF
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