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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