Nuprl Lemma : orderedpair-second
∀a,b:coSet{i:l}. seteq(snds((a,b));{b})
Proof
Definitions occuring in Statement :
orderedpair-snds: snds(pr),
orderedpairset: (a,b),
singleset: {a},
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
or: P ∨ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
orderedpairset: (a,b),
guard: {T},
set-predicate: set-predicate{i:l}(s;a.P[a]),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
orderedpair-snds: snds(pr),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
setmem-pairset,
or_wf,
pairset_wf,
setmem-unionset,
setmem_functionality_1,
seteq-orderedpairs-iff,
seteq_inversion,
iff_wf,
sub-set_wf,
setmem-singleset,
orderedpairset_functionality,
seteq_weakening,
seteq_functionality,
setmem_wf,
orderedpair-fst_wf,
seteq_wf,
unionset_wf,
setmem-sub-coset,
fst-orderedpairset,
coSet_wf,
singleset_wf,
orderedpairset_wf,
orderedpair-snds_wf,
co-seteq-iff
Rules used in proof :
inrFormation,
dependent_pairFormation,
andLevelFunctionality,
productEquality,
cumulativity,
setEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
impliesFunctionality,
independent_pairFormation,
addLevel,
because_Cache,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b:coSet\{i:l\}. seteq(snds((a,b));\{b\})
Date html generated:
2018_07_29-AM-10_02_22
Last ObjectModification:
2018_07_18-PM-03_07_15
Theory : constructive!set!theory
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