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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-inf2 9101* 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 9102 shows it converted to abbreviations. This axiom was derived as theorem axinf2 9100 above, using our version of Infinity ax-inf 9098 and the Axiom of Regularity ax-reg 9053. We will reference ax-inf2 9101 instead of axinf2 9100 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9098 from ax-inf2 9101 is shown by theorem axinf 9104. (Contributed by NM, 3-Nov-1996.)
𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Theoremzfinf2 9102* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9101 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 9103 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 9081.

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 7585 and Fin = V (the universe of all sets) by fineqv 8730. 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 7595 through peano5 7599 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

ω ∈ V

Theoremaxinf 9104* The first version of the Axiom of Infinity ax-inf 9098 proved from the second version ax-inf2 9101. Note that we didn't use ax-reg 9053, unlike the other direction axinf2 9100. (Contributed by NM, 24-Apr-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf5 9105 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 9097). 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 9106 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
ω ∈ On

Theoremdfom3 9107* The class of natural numbers ω can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), 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 9108* A simplification of elom 7577 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
(𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥𝐴𝑥))

Theoremdfom4 9109* A simplification of df-om 7575 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}

Theoremdfom5 9110 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 9111 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 9112 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 9113 A finite set is countable (weaker version of isfinite 9112). (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝐴 ∈ Fin → 𝐴 ≼ ω)

Theoremnnsdom 9114 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 9115 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 9116 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
ω ≈ suc ω

Theoreminfdifsn 9117 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 9118 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 9119* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 8771 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremnoinfep 9120* Using the Axiom of Regularity in the form zfregfr 9065, 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 9121 Extend class notation with the Cantor normal form function.
class CNF

Definitiondf-cnf 9122* 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 9155 of this function in terms of df-oi 8971. (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 9123* 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 9124* 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 9125* Lemma for cantnf 9153. The function appearing in cantnfval 9128 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
𝐹 = seqω((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)), ∅)       𝐹:ω⟶On

Theoremcantnfs 9126 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 9127 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 9128* 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 9129* 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 9130* 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 9131* 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 9132* 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 9133 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 9134 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 9135 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 9136* 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 9137* 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 9138* Lemma for cantnfp1 9141. (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 9139* Lemma for cantnfp1 9141. (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 9140* Lemma for cantnfp1 9141. (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 9141* 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 9142* The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9014). (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 9143* 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 9144* 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 9145* Lemma for cantnf 9153. (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 9146* Lemma for cantnf 9153. (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 9147* Lemma for cantnf 9153. (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 9148* Lemma for cantnf 9153. (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 9149* Lemma for cantnf 9153. 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 9150* Lemma for cantnf 9153. (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 9151* Lemma for cantnf 9153. 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 8225 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 9141 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 9152* Lemma for cantnf 9153. Complete the induction step of cantnflem3 9151. (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 9153* 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 indices 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 9137, 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 9154* 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 9155* An alternate definition of df-cnf 9122 which relies on cantnf 9153. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9124 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 9156 Simplify the isomorphism of cantnf 9153 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 9157* Construct lexicographic order on a function space based on a reverse well-ordering of the indices 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 9158* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7051.) (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 9159* Lemma for cnfcom 9160. (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 9160* 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 9161* Lemma for cnfcom2 9162. (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 9162* 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 9163* Lemma for cnfcom3 9164. (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 9164* 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 9166.) (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 9165* Lemma for cnfcom3c 9166. (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 9166* Wrap the construction of cnfcom3 9164 into an existential 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 9167* 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 9168 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 9168* Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9167 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 9169* Alternate expression for the existence of transitive closures tz9.1 9168: 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 9170* The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9171. (Contributed by Mario Carneiro, 22-Mar-2013.)
(( E Fr 𝐴𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)

Theoremzfregs 9171* 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 9170. (Contributed by NM, 17-Sep-2003.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)

Theoremzfregs2 9172* 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 9173* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)

Theoremsetind2 9174 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 9175 Extend class notation to include the transitive closure function.
class TC

Definitiondf-tc 9176* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
TC = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ Tr 𝑦)})

Theoremtcvalg 9177* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9098; see tz9.1 9168.) (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})

Theoremtcid 9178 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 9179 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Tr (TC‘𝐴)

Theoremtcmin 9180 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))

Theoremtc2 9181* A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}

Theoremtcsni 9182 The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
𝐴 ∈ V       (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})

Theoremtcss 9183 The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Theoremtcel 9184 The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Theoremtcidm 9185 The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
(TC‘(TC‘𝐴)) = (TC‘𝐴)

Theoremtc0 9186 The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
(TC‘∅) = ∅

Theoremtc00 9187 The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
(𝐴𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅))

2.6.5  Rank

Syntaxcr1 9188 Extend class definition to include the cumulative hierarchy of sets function.
class 𝑅1

Syntaxcrnk 9189 Extend class definition to include rank function.
class rank

Definitiondf-r1 9190 Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (𝑅1). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 9217). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 9194, r1suc 9196, and r1lim 9198. Theorem r1val1 9212 shows a recursive definition that works for all values, and theorems r1val2 9263 and r1val3 9264 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), V with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)
𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)

Definitiondf-rank 9191* Define the rank function. See rankval 9242, rankval2 9244, rankval3 9266, or rankval4 9293 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 𝑅1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 9259 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 9262. (Contributed by NM, 11-Oct-2003.)
rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})

Theoremr1funlim 9192 The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9193 avoids ax-rep 5176.) (Contributed by Mario Carneiro, 16-Nov-2014.)
(Fun 𝑅1 ∧ Lim dom 𝑅1)

Theoremr1fnon 9193 The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
𝑅1 Fn On

Theoremr10 9194 Value of the cumulative hierarchy of sets function at . Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
(𝑅1‘∅) = ∅

Theoremr1sucg 9195 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))

Theoremr1suc 9196 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
(𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))

Theoremr1limg 9197* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))

Theoremr1lim 9198* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
((𝐴𝐵 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))

Theoremr1fin 9199 The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
(𝐴 ∈ ω → (𝑅1𝐴) ∈ Fin)

Theoremr1sdom 9200 Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)
((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45263
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