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Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

2.5.2  Axiom of Infinity equivalents

Theoreminf0 8401* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 8418. (Contributed by NM, 15-Oct-1996.)
ω ∈ V       𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf1 8402 Variation of Axiom of Infinity (using zfinf 8419 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))

Theoreminf2 8403* Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8419 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)

Theoreminf3lema 8404* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))

Theoreminf3lemb 8405* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐹‘∅) = ∅

Theoreminf3lemc 8406* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹𝐴)))

Theoreminf3lemd 8407* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)

Theoreminf3lem1 8408* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))

Theoreminf3lem2 8409* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ 𝑥))

Theoreminf3lem3 8410* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8383. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ (𝐹‘suc 𝐴)))

Theoreminf3lem4 8411* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ⊊ (𝐹‘suc 𝐴)))

Theoreminf3lem5 8412* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝐹𝐵) ⊊ (𝐹𝐴)))

Theoreminf3lem6 8413* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

Theoreminf3lem7 8414* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7029. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)

Theoreminf3 8415 Our Axiom of Infinity ax-inf 8418 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8403, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 8420 and zfinf2 8422.) The main proof is provided by inf3lema 8404 through inf3lem7 8414, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8414. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.
(As posted to sci.logic on 30-Oct-1996, with annotations added.)

Theorem:  The statement "There exists a nonempty set that is a subset
of its union" implies the Axiom of Infinity.

Proof:  Let X be a nonempty set which is a subset of its union; the
latter
property is equivalent to saying that for any y in X, there exists a z
in X
such that y is in z.

Define by finite recursion a function F:omega-->(power X) such that
F_0 = 0  (See inf3lemb 8405.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8406.)
Note: ^ means intersect, < means \in ("element of").
(Finite recursion as typically done requires the existence of omega;
to avoid this we can just use transfinite recursion restricted to omega.
F is a class-term that is not necessarily a set at this point.)

Lemma 1.  F_n subset F_n+1.  (See inf3lem1 8408.)
Proof:  By induction:  F_0 subset F_1.  If y < F_n+1, then y^X subset
F_n,
so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2.

Lemma 2.  F_n =/= X.  (See inf3lem2 8409.)
Proof:  By induction:  F_0 =/= X because X is not empty.  Assume F_n =/=
X.
Then there is a y in X that is not in F_n.  By definition of X, there is
a
z in X that contains y.  Suppose F_n+1 = X.  Then z is in F_n+1, and z^X
contains y, so z^X is not a subset of F_n, contrary to the definition of
F_n+1.

Lemma 3.  F_n =/= F_n+1.  (See inf3lem3 8410.)
Proof:  Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have
F_n+1 = {y<X | y^(X-F_n) = 0}.  Let q = {y<X-F_n | y^(X-F_n) = 0}.
Then q subset F_n+1.  Since X-F_n is not empty by Lemma 2 and q is the
set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q
and therefore F_n+1 have an element not in F_n.

Lemma 4.  F_n proper_subset F_n+1.  (See inf3lem4 8411.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8412.)
Proof:  Fix m and use induction on n > m.  Basis: F_m proper_subset
F_m+1
by Lemma 4.  Induction:  Assume F_m proper_subset F_n.  Then since F_n
proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper
subset.

By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1.  (See inf3lem6 8413.)
Thus, the inverse of F is a function with range omega and domain a
subset
of power X, so omega exists by Replacement.  (See inf3lem7 8414.)
Q.E.D.

(Contributed by NM, 29-Oct-1996.)
𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)       ω ∈ V

Theoreminfeq5i 8416 Half of infeq5 8417. (Contributed by Mario Carneiro, 16-Nov-2014.)
(ω ∈ V → ∃𝑥 𝑥 𝑥)

Theoreminfeq5 8417 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 8423.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
(∃𝑥 𝑥 𝑥 ↔ ω ∈ V)

2.6  ZF Set Theory - add the Axiom of Infinity

2.6.1  Introduce the Axiom of Infinity

Axiomax-inf 8418* Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set 𝑥, an infinite set 𝑦 built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 8402 and inf2 8403). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8422 and omex 8423 and are based on the (nontrivial) proof of inf3 8415. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8421. Theorem inf0 8401 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8425 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8421 requires this axiom along with Regularity ax-reg 8380 for its derivation (as theorem axinf2 8420 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8421 instead of this one. The derivation of this axiom from ax-inf2 8421 is shown by theorem axinf 8424.

Proofs should normally use the standard version ax-inf2 8421 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfinf 8419* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))

Theoremaxinf2 8420* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8418 and Regularity ax-reg 8380.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8421 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Axiomax-inf2 8421* A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 8422 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8420 above, using our version of Infinity ax-inf 8418 and the Axiom of Regularity ax-reg 8380. We will reference ax-inf2 8421 instead of axinf2 8420 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8418 from ax-inf2 8421 is shown by theorem axinf 8424. (Contributed by NM, 3-Nov-1996.)
𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Theoremzfinf2 8422* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8421 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)

2.6.2  Existence of omega (the set of natural numbers)

Theoremomex 8423 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 8401.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 6968 and Fin = V (the universe of all sets) by fineqv 8060. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 6977 through peano5 6981 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

ω ∈ V

Theoremaxinf 8424* The first version of the Axiom of Infinity ax-inf 8418 proved from the second version ax-inf2 8421. Note that we didn't use ax-reg 8380, unlike the other direction axinf2 8420. (Contributed by NM, 24-Apr-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf5 8425 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 8417). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)
𝑥 𝑥 𝑥

Theoremomelon 8426 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
ω ∈ On

Theoremdfom3 8427* The class of natural numbers omega can be defined as the smallest "inductive set," which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.)
ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}

Theoremelom3 8428* A simplification of elom 6960 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
(𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥𝐴𝑥))

Theoremdfom4 8429* A simplification of df-om 6958 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}

Theoremdfom5 8430 ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
ω = {𝑥 ∣ Lim 𝑥}

Theoremoancom 8431 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
(1𝑜 +𝑜 ω) ≠ (ω +𝑜 1𝑜)

Theoremisfinite 8432 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)
(𝐴 ∈ Fin ↔ 𝐴 ≺ ω)

Theoremfict 8433 A finite set is countable (weaker version of isfinite 8432). (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝐴 ∈ Fin → 𝐴 ≼ ω)

Theoremnnsdom 8434 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)
(𝐴 ∈ ω → 𝐴 ≺ ω)

Theoremomenps 8435 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)
ω ≈ (ω ∖ {∅})

Theoremomensuc 8436 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
ω ≈ suc ω

Theoreminfdifsn 8437 Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
(ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Theoreminfdiffi 8438 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)

Theoremunbnn3 8439* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 8101 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremnoinfep 8440* Using the Axiom of Regularity in the form zfregfr 8392, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)

2.6.3  Cantor normal form

Syntaxccnf 8441 Extend class notation with the Cantor normal form function.
class CNF

Definitiondf-cnf 8442* Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥𝑜 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑𝑜 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 8475 of this function in terms of df-oi 8298. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))

Theoremcantnffval 8443* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))

Theoremcantnfdm 8444* The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)

Theoremcantnfvalf 8445* Lemma for cantnf 8473. The function appearing in cantnfval 8448 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
𝐹 = seq𝜔((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)), ∅)       𝐹:ω⟶On

Theoremcantnfs 8446 Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))

Theoremcantnfcl 8447 Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)       (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))

Theoremcantnfval 8448* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

Theoremcantnfval2 8449* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))

Theoremcantnfsuc 8450* The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)       ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))

Theoremcantnfle 8451* A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴𝑜 𝑥) ·𝑜 (𝐹𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)    &   (𝜑𝐶𝐵)       (𝜑 → ((𝐴𝑜 𝐶) ·𝑜 (𝐹𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))

Theoremcantnflt 8452* An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴𝑜 𝐶 where 𝐶 is larger than any exponent (𝐺𝑥), 𝑥𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   (𝜑𝐹𝑆)    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   (𝜑𝐾 ∈ suc dom 𝐺)    &   (𝜑𝐶 ∈ On)    &   (𝜑 → (𝐺𝐾) ⊆ 𝐶)       (𝜑 → (𝐻𝐾) ∈ (𝐴𝑜 𝐶))

Theoremcantnflt2 8453 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐹𝑆)    &   (𝜑 → ∅ ∈ 𝐴)    &   (𝜑𝐶 ∈ On)    &   (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴𝑜 𝐶))

Theoremcantnff 8454 The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴𝑜 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))

Theoremcantnf0 8455 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑 → ∅ ∈ 𝐴)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)

Theoremcantnfrescl 8456* A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑𝐵𝐷)    &   ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   𝑇 = dom (𝐴 CNF 𝐷)       (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))

Theoremcantnfres 8457* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑𝐵𝐷)    &   ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)    &   (𝜑 → ∅ ∈ 𝐴)    &   𝑇 = dom (𝐴 CNF 𝐷)    &   (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))

Theoremcantnfp1lem1 8458* Lemma for cantnfp1 8461. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑𝐹𝑆)

Theoremcantnfp1lem2 8459* Lemma for cantnfp1 8461. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))    &   (𝜑 → ∅ ∈ 𝑌)    &   𝑂 = OrdIso( E , (𝐹 supp ∅))       (𝜑 → dom 𝑂 = suc dom 𝑂)

Theoremcantnfp1lem3 8460* Lemma for cantnfp1 8461. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))    &   (𝜑 → ∅ ∈ 𝑌)    &   𝑂 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐹‘(𝑂𝑘))) +𝑜 𝑧)), ∅)    &   𝐾 = OrdIso( E , (𝐺 supp ∅))    &   𝑀 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐾𝑘)) ·𝑜 (𝐺‘(𝐾𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))

Theoremcantnfp1 8461* If 𝐹 is created by adding a single term (𝐹𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐺𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐴)    &   (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))

Theoremoemapso 8462* The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 8341). (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑𝑇 Or 𝑆)

Theoremoemapval 8463* Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)       (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))

Theoremoemapvali 8464* If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}       (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))

Theoremcantnflem1a 8465* Lemma for cantnf 8473. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}       (𝜑𝑋 ∈ (𝐺 supp ∅))

Theoremcantnflem1b 8466* Lemma for cantnf 8473. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))       ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))

Theoremcantnflem1c 8467* Lemma for cantnf 8473. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))       ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) ∧ 𝑥𝐵) ∧ ((𝐹𝑥) ≠ ∅ ∧ (𝑂𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))

Theoremcantnflem1d 8468* Lemma for cantnf 8473. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥𝐵 ↦ if(𝑥𝑋, (𝐹𝑥), ∅))) ∈ (𝐻‘suc (𝑂𝑋)))

Theoremcantnflem1 8469* Lemma for cantnf 8473. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐹𝑆)    &   (𝜑𝐺𝑆)    &   (𝜑𝐹𝑇𝐺)    &   𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}    &   𝑂 = OrdIso( E , (𝐺 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑂𝑘)) ·𝑜 (𝐺‘(𝑂𝑘))) +𝑜 𝑧)), ∅)       (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺))

Theoremcantnflem2 8470* Lemma for cantnf 8473. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)       (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))

Theoremcantnflem3 8471* Lemma for cantnf 8473. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 7570 to factor 𝐶 into the form ((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴𝑜 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴𝑜 𝑋) ≤ (𝐴𝑜 𝑋) ·𝑜 𝑌𝐶, 𝑍 has a normal form, and by appending the term (𝐴𝑜 𝑋) ·𝑜 𝑌 using cantnfp1 8461 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)    &   𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}    &   𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)    &   (𝜑𝐺𝑆)    &   (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)    &   𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))       (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))

Theoremcantnflem4 8472* Lemma for cantnf 8473. Complete the induction step of cantnflem3 8471. (Contributed by Mario Carneiro, 25-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐶 ∈ (𝐴𝑜 𝐵))    &   (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   (𝜑 → ∅ ∈ 𝐶)    &   𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}    &   𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)       (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))

Theoremcantnf 8473* The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴𝑜 𝑓(𝑎1)) ∘ 𝑎1) +𝑜 ((𝐴𝑜 𝑓(𝑎2)) ∘ 𝑎2) +𝑜 ... over all indexes 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴𝑜 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 8457, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))

Theoremoemapwe 8474* The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴𝑜 𝐵)))

Theoremcantnffval2 8475* An alternate definition of df-cnf 8442 which relies on cantnf 8473. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8444 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}       (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))

Theoremcantnff1o 8476 Simplify the isomorphism of cantnf 8473 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
𝑆 = dom (𝐴 CNF 𝐵)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)       (𝜑 → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))

Theoremwemapwe 8477* Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑆 We 𝐵)    &   (𝜑𝐵 ≠ ∅)    &   𝐹 = OrdIso(𝑅, 𝐴)    &   𝐺 = OrdIso(𝑆, 𝐵)    &   𝑍 = (𝐺‘∅)       (𝜑𝑇 We 𝑈)

Theoremoef1o 8478* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6457.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
(𝜑𝐹:𝐴1-1-onto𝐶)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ (On ∖ 1𝑜))    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐶 ∈ On)    &   (𝜑𝐷 ∈ On)    &   (𝜑 → (𝐹‘∅) = ∅)    &   𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))    &   𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))       (𝜑𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))

Theoremcnfcomlem 8479* Lemma for cnfcom 8480. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   (𝜑𝐼 ∈ dom 𝐺)    &   (𝜑𝑂 ∈ (ω ↑𝑜 (𝐺𝐼)))    &   (𝜑 → (𝑇𝐼):(𝐻𝐼)–1-1-onto𝑂)       (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))

Theoremcnfcom 8480* Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   (𝜑𝐼 ∈ dom 𝐺)       (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))

Theoremcnfcom2lem 8481* Lemma for cnfcom2 8482. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ∅ ∈ 𝐵)       (𝜑 → dom 𝐺 = suc dom 𝐺)

Theoremcnfcom2 8482* Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ∅ ∈ 𝐵)       (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))

Theoremcnfcom3lem 8483* Lemma for cnfcom3 8484. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ω ⊆ 𝐵)       (𝜑𝑊 ∈ (On ∖ 1𝑜))

Theoremcnfcom3 8484* Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 8486.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))    &   𝐹 = ((ω CNF 𝐴)‘𝐵)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   (𝜑 → ω ⊆ 𝐵)    &   𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))    &   𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))    &   𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))       (𝜑𝑁:𝐵1-1-onto→(ω ↑𝑜 𝑊))

Theoremcnfcom3clem 8485* Lemma for cnfcom3c 8486. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
𝑆 = dom (ω CNF 𝐴)    &   𝐹 = ((ω CNF 𝐴)‘𝑏)    &   𝐺 = OrdIso( E , (𝐹 supp ∅))    &   𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)    &   𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))    &   𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))    &   𝑊 = (𝐺 dom 𝐺)    &   𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))    &   𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))    &   𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))    &   𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)       (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))

Theoremcnfcom3c 8486* Wrap the construction of cnfcom3 8484 into an existence quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
(𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))

2.6.4  Transitive closure

Theoremtrcl 8487* For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 8488 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐴 ∈ V    &   𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)    &   𝐶 = 𝑦 ∈ ω (𝐹𝑦)       (𝐴𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥))

Theoremtz9.1 8488* Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8487 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself?

(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)

𝐴 ∈ V       𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))

Theoremtz9.1c 8489* Alternate expression for the existence of transitive closures tz9.1 8488: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V        {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V

Theoremepfrs 8490* The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 8491. (Contributed by Mario Carneiro, 22-Mar-2013.)
(( E Fr 𝐴𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)

Theoremzfregs 8491* The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 8490. (Contributed by NM, 17-Sep-2003.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)

Theoremzfregs2 8492* Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))

Theoremsetind 8493* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)

Theoremsetind2 8494 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
(𝒫 𝐴𝐴𝐴 = V)

Syntaxctc 8495 Extend class notation to include the transitive closure function.
class TC

Definitiondf-tc 8496* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
TC = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ Tr 𝑦)})

Theoremtcvalg 8497* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 8418; see tz9.1 8488.) (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})

Theoremtcid 8498 Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉𝐴 ⊆ (TC‘𝐴))

Theoremtctr 8499 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Tr (TC‘𝐴)

Theoremtcmin 8500 Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

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