Nuprl Lemma : Sierpinski-equal
∀[x,y:Sierpinski]. uiff(x = y ∈ Sierpinski;x = ⊥ ∈ Sierpinski ⇐⇒ y = ⊥ ∈ Sierpinski)
Proof
Definitions occuring in Statement :
Sierpinski: Sierpinski,
Sierpinski-bottom: ⊥,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
Sierpinski: Sierpinski,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
guard: {T},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal-wf-T-base,
Sierpinski_wf,
equal_wf,
quotient-member-eq,
nat_wf,
bool_wf,
iff_wf,
two-class-equiv-rel,
Sierpinski-bottom_wf,
Sierpinski-equal-bottom,
iff_imp_equal_bool,
assert_functionality_wrt_uiff,
assert_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
baseClosed,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
functionEquality,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
functionExtensionality,
applyEquality,
productEquality,
isect_memberEquality
Latex:
\mforall{}[x,y:Sierpinski]. uiff(x = y;x = \mbot{} \mLeftarrow{}{}\mRightarrow{} y = \mbot{})
Date html generated:
2019_10_31-AM-06_36_27
Last ObjectModification:
2017_07_28-AM-09_12_14
Theory : synthetic!topology
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