Nuprl Lemma : setmemfunclemma
∀x1,s1,x2,s2:coSet{i:l}. (seteq(x1;x2) ⇒ seteq(s1;s2) ⇒ {(x1 ∈ s1) ⇐⇒ (x2 ∈ s2)})
Proof
Definitions occuring in Statement :
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
set-item_wf,
seteq_transitivity,
seteq_inversion,
setmem-iff,
coSet_wf,
seteq_wf,
setmem_wf,
co-seteq-iff
Rules used in proof :
dependent_pairFormation,
isectElimination,
independent_pairFormation,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x1,s1,x2,s2:coSet\{i:l\}. (seteq(x1;x2) {}\mRightarrow{} seteq(s1;s2) {}\mRightarrow{} \{(x1 \mmember{} s1) \mLeftarrow{}{}\mRightarrow{} (x2 \mmember{} s2)\})
Date html generated:
2018_07_29-AM-09_51_29
Last ObjectModification:
2018_07_11-PM-03_07_17
Theory : constructive!set!theory
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