Nuprl Lemma : Sierpinski-unequal-1
⊥ = ⊤ ∈ (ℕ ⟶ 𝔹) ⇐⇒ False
Proof
Definitions occuring in Statement :
Sierpinski-top: ⊤,
Sierpinski-bottom: ⊥,
nat: ℕ,
bool: 𝔹,
iff: P ⇐⇒ Q,
false: False,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
Sierpinski-top: ⊤,
Sierpinski-bottom: ⊥,
rev_implies: P ⇐ Q
Lemmas referenced :
Sierpinski-top_wf,
Sierpinski-bottom_wf,
bool_wf,
nat_wf,
equal_wf,
btrue_neq_bfalse,
le_wf,
false_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
cut,
applyEquality,
lambdaEquality,
hypothesisEquality,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
because_Cache,
hypothesis,
unionElimination,
isectElimination,
independent_isectElimination,
dependent_pairFormation,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
independent_functionElimination,
functionEquality
Latex:
\mbot{} = \mtop{} \mLeftarrow{}{}\mRightarrow{} False
Date html generated:
2019_10_31-AM-06_35_21
Last ObjectModification:
2016_01_17-AM-09_35_56
Theory : synthetic!topology
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