Nuprl Lemma : equipollent-sets
∀[T:Type]. ∀[P,Q:T ⟶ ℙ]. {x:T| P[x]} ~ {x:T| Q[x]} supposing ∀x:T. (P[x] ⇐⇒ Q[x])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
ext-eq: A ≡ B,
and: P ∧ Q,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent_weakening_ext-eq,
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
sqequalRule,
universeEquality,
because_Cache,
independent_isectElimination,
functionEquality,
independent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P,Q:T {}\mrightarrow{} \mBbbP{}]. \{x:T| P[x]\} \msim{} \{x:T| Q[x]\} supposing \mforall{}x:T. (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x])
Date html generated:
2016_10_21-AM-10_58_12
Last ObjectModification:
2016_08_07-PM-11_36_41
Theory : equipollence!!cardinality!
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