Nuprl Lemma : orderedpair-first
∀a,b:coSet{i:l}. seteq(orderedpair-fsts((a,b));{a})
Proof
Definitions occuring in Statement :
orderedpair-fsts: orderedpair-fsts(pr),
orderedpairset: (a,b),
singleset: {a},
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
cand: A c∧ B,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
prop: ℙ,
orderedpairset: (a,b),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
orderedpair-fsts: orderedpair-fsts(pr)
Lemmas referenced :
seteq-iff-setsubset,
seteq_weakening,
setmem_functionality,
setmem-intersectionset,
all_wf,
or_wf,
intersectionset_wf,
setsubset-iff,
setmem-pairset,
iff_wf,
setmem-singleset,
seteq_wf,
co-seteq-iff,
pairset_wf,
orderedpairset_wf,
setmem_wf,
singleset_wf,
coSet_wf
Rules used in proof :
impliesLevelFunctionality,
allLevelFunctionality,
unionElimination,
functionEquality,
lambdaEquality,
instantiate,
cumulativity,
allFunctionality,
impliesFunctionality,
addLevel,
independent_pairFormation,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inlFormation,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
dependent_pairFormation,
hypothesis,
extract_by_obid,
introduction,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}a,b:coSet\{i:l\}. seteq(orderedpair-fsts((a,b));\{a\})
Date html generated:
2018_07_29-AM-10_01_58
Last ObjectModification:
2018_07_18-PM-03_00_42
Theory : constructive!set!theory
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