Nuprl Lemma : assert_of_eq_pair
∀[s,t:DSet]. ∀[a,b:|s| × |t|]. uiff(↑(a =b b);a = b ∈ (|s| × |t|))
Proof
Definitions occuring in Statement :
eq_pair: a =b b,
dset: DSet,
set_car: |p|,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eq_pair: a =b b,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
dset: DSet,
pi1: fst(t),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
bfalse: ff,
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
assert_wf,
eq_pair_wf,
assert_witness,
equal_wf,
set_car_wf,
dset_wf,
iff_transitivity,
infix_ap_wf,
bool_wf,
set_eq_wf,
eqtt_to_assert,
assert_of_dset_eq,
iff_weakening_uiff,
assert_of_band,
pi1_wf,
pi2_wf,
uiff_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
setElimination,
rename,
because_Cache,
addLevel,
independent_pairFormation,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_functionElimination,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality
Latex:
\mforall{}[s,t:DSet]. \mforall{}[a,b:|s| \mtimes{} |t|]. uiff(\muparrow{}(a =\msubb{} b);a = b)
Date html generated:
2017_10_01-AM-08_13_18
Last ObjectModification:
2017_02_28-PM-01_58_02
Theory : sets_1
Home
Index