Theorem List for New Foundations Explorer - 6301-6329 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nchoicelem15 6301 |
Lemma for nchoice 6306. When the special set generator does not
yield a
singleton, then the cardinal is raisable. (Contributed by SF,
19-Mar-2015.)
|
⊢ ((M ∈ NC ∧ 1c <c Nc ( Spac
‘M)) → (M ↑c 0c)
∈ NC
) |
|
Theorem | nchoicelem16 6302* |
Lemma for nchoice 6306. Set up stratification for nchoicelem17 6303.
(Contributed by SF, 19-Mar-2015.)
|
⊢ {t ∣ ( ≤c We NC → ∀m ∈ NC ( Nc ( Spac
‘m) = (1c
+c t) → (( Spac ‘ Tc m)
∈ Fin ∧ ( Nc ( Spac ‘ Tc m) =
( Tc Nc (
Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) =
( Tc Nc (
Spac ‘m) +c 2c)))))}
∈ V |
|
Theorem | nchoicelem17 6303 |
Lemma for nchoice 6306. If the special set of a cardinal is finite,
then
so is the special set of its T-raising, and there is a calculable
relationship between their sizes. Theorem 7.2 of [Specker] p. 974.
(Contributed by SF, 19-Mar-2015.)
|
⊢ (( ≤c We NC ∧ M ∈ NC ∧ ( Spac
‘M) ∈ Fin ) → ((
Spac ‘ Tc M)
∈ Fin ∧ ( Nc ( Spac ‘ Tc M) =
( Tc Nc (
Spac ‘M) +c 1c) ∨ Nc ( Spac ‘ Tc M) =
( Tc Nc (
Spac ‘M) +c
2c)))) |
|
Theorem | nchoicelem18 6304 |
Lemma for nchoice 6306. Set up stratification for nchoicelem19 6305.
(Contributed by SF, 20-Mar-2015.)
|
⊢ {x ∣ ( Spac
‘x) ∈ Fin } ∈ V |
|
Theorem | nchoicelem19 6305 |
Lemma for nchoice 6306. Assuming well-ordering, there is a cardinal
with
a finite special set that is its own T-raising. Theorem 7.3 of
[Specker] p. 974. (Contributed by SF,
20-Mar-2015.)
|
⊢ ( ≤c We NC → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m =
m)) |
|
Theorem | nchoice 6306 |
Cardinal less than or equal does not well-order the cardinals. This is
equivalent to saying that the axiom of choice from ZFC is false in NF.
Theorem 7.5 of [Specker] p. 974.
(Contributed by SF, 20-Mar-2015.)
|
⊢ ¬ ≤c We NC |
|
2.4.7 Finite recursion
|
|
Syntax | cfrec 6307 |
Extend the definition of a class to include the finite recursive function
generator.
|
class
FRec (F, I) |
|
Definition | df-frec 6308* |
Define the finite recursive function generator. This is a function over
Nn that obeys the standard
recursion relationship. Definition
adapted from theorem XI.3.24 of [Rosser]
p. 412. (Contributed by Scott
Fenton, 30-Jul-2019.)
|
⊢ FRec (F, I) = Clos1 ({〈0c, I〉}, PProd ((x ∈ V ↦ (x +c 1c)),
F)) |
|
Theorem | freceq12 6309 |
Equality theorem for finite recursive function generator. (Contributed
by Scott Fenton, 31-Jul-2019.)
|
⊢ ((F =
G ∧
I = J)
→ FRec (F, I) = FRec (G, J)) |
|
Theorem | frecexg 6310 |
The finite recursive function generator preserves sethood. (Contributed
by Scott Fenton, 30-Jul-2019.)
|
⊢ F = FRec (G, I) ⇒ ⊢ (G ∈ V →
F ∈
V) |
|
Theorem | frecex 6311 |
The finite recursive function generator preserves sethood. (Contributed
by Scott Fenton, 30-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ G ∈ V ⇒ ⊢ F ∈ V |
|
Theorem | frecxp 6312 |
Subset relationship for the finite recursive function generator.
(Contributed by Scott Fenton, 30-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ G ∈ V ⇒ ⊢ F ⊆ ( Nn × (ran
G ∪ {I})) |
|
Theorem | frecxpg 6313 |
Subset relationship for the finite recursive function generator.
(Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I) ⇒ ⊢ (G ∈ V →
F ⊆ (
Nn × (ran G ∪ {I}))) |
|
Theorem | dmfrec 6314 |
The domain of the finite recursive function generator is the naturals.
(Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ V)
& ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G) ⇒ ⊢ (φ
→ dom F = Nn ) |
|
Theorem | fnfreclem1 6315* |
Lemma for fnfrec 6318. Establish stratification for induction.
(Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ (F ∈ V →
{w ∣
∀y∀z((wFy ∧ wFz) →
y = z)} ∈
V) |
|
Theorem | fnfreclem2 6316 |
Lemma for fnfrec 6318. Calculate the unique value of F at zero.
(Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ V)
& ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G) ⇒ ⊢ (φ
→ (0cFX → X =
I)) |
|
Theorem | fnfreclem3 6317* |
Lemma for fnfrec 6318. The value of F at a successor is G related
to a previous element. (Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ V)
& ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G)
& ⊢ (φ
→ X ∈ Nn ) & ⊢ (φ
→ (X +c
1c)FY) ⇒ ⊢ (φ
→ ∃z(XFz ∧ zGY)) |
|
Theorem | fnfrec 6318 |
The recursive function generator is a function over the finite
cardinals. (Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ Funs ) & ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G) ⇒ ⊢ (φ
→ F Fn Nn ) |
|
Theorem | frec0 6319 |
Calculate the value of the finite recursive function generator at zero.
(Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ Funs ) & ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G) ⇒ ⊢ (φ
→ (F ‘0c) =
I) |
|
Theorem | frecsuc 6320 |
Calculate the value of the finite recursive function generator at a
successor. (Contributed by Scott Fenton, 31-Jul-2019.)
|
⊢ F = FRec (G, I)
& ⊢ (φ
→ G ∈ Funs ) & ⊢ (φ
→ I ∈ dom G)
& ⊢ (φ
→ ran G ⊆ dom G)
& ⊢ (φ
→ X ∈ Nn
) ⇒ ⊢ (φ
→ (F ‘(X +c 1c)) =
(G ‘(F ‘X))) |
|
2.5 Cantorian and Strongly Cantorian
Sets
|
|
Syntax | ccan 6321 |
Extend the definition of class to include the class of all Cantorian
sets.
|
class
Can |
|
Definition | df-can 6322 |
Define the class of all Cantorian sets. These are so-called because
Cantor's Theorem A~<℘A
holds for these sets. (Contributed by
Scott Fenton, 19-Apr-2021.)
|
⊢ Can = {x ∣ ℘1x ≈ x} |
|
Syntax | cscan 6323 |
Extend the definition of class to include the class of all strongly
Cantorian sets.
|
class
SCan |
|
Definition | df-scan 6324* |
Define the class of strongly Cantorian sets. Unline general Cantorian
sets, this fixes a specific mapping between x and ℘1x.
(Contributed by Scott Fenton, 19-Apr-2021.)
|
⊢ SCan = {x ∣ (y ∈ x ↦ {y}) ∈
V} |
|
Theorem | dmsnfn 6325* |
The domain of the singleton function. (Contributed by Scott Fenton,
20-Apr-2021.)
|
⊢ dom (x
∈ A
↦ {x})
= A |
|
Theorem | epelcres 6326 |
Version of epelc 4765 with a restriction in place. (Contributed by
Scott Fenton, 20-Apr-2021.)
|
⊢ Y ∈ V ⇒ ⊢ (X ∈ A →
(X( E ↾
A)Y
↔ X ∈ Y)) |
|
Theorem | elcan 6327 |
Membership in the class of Cantorian sets. (Contributed by Scott
Fenton, 19-Apr-2021.)
|
⊢ (A ∈ Can ↔ ℘1A ≈ A) |
|
Theorem | elscan 6328* |
Membership in the class of strongly Cantorian sets. (Contributed by
Scott Fenton, 19-Apr-2021.)
|
⊢ (A ∈ SCan ↔
(x ∈
A ↦
{x}) ∈
V) |
|
Theorem | scancan 6329 |
Strongly Cantorian implies Cantorian. (Contributed by Scott Fenton,
19-Apr-2021.)
|
⊢ (A ∈ SCan →
A ∈
Can ) |