Nuprl Lemma : union_functionality_wrt_equipollent
∀[A,B,C,D:Type]. (A ~ B ⇒ C ~ D ⇒ A + C ~ B + D)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
isl: isl(x),
not: ¬A,
false: False,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent_wf,
equal_wf,
biject_wf,
btrue_wf,
bfalse_wf,
and_wf,
isl_wf,
btrue_neq_bfalse,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_pairFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqequalRule,
inlEquality,
applyEquality,
inrEquality,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
applyLambdaEquality,
dependent_set_memberEquality,
setElimination,
voidElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[A,B,C,D:Type]. (A \msim{} B {}\mRightarrow{} C \msim{} D {}\mRightarrow{} A + C \msim{} B + D)
Date html generated:
2019_06_20-PM-02_16_52
Last ObjectModification:
2019_06_19-PM-06_19_59
Theory : equipollence!!cardinality!
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