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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremen2lp 8801 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
¬ (𝐴𝐵𝐵𝐴)

Theoremcnvepnep 8802 The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 8801. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.)
( E ∩ E ) = ∅

Theoremepnsym 8803 The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
E ≠ E

Theoremelnotel 8804 A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theoremelnel 8805 A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
(𝐴𝐵𝐵𝐴)

Theoremen3lplem1 8806* Lemma for en3lp 8808. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lplem2 8807* Lemma for en3lp 8808. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lp 8808 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 40024 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

Theorempreleqg 8809 Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 8810. (Contributed by AV, 15-Jun-2022.)
(((𝐴𝐵𝐵𝑉𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Theorempreleq 8810 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Revised by AV, 15-Jun-2022.)
𝐵 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

TheorempreleqALT 8811 Alternate proof of preleq 8810, not based on preleqg 8809: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V    &   𝐷 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremopthreg 8812 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8788 (via the preleq 8810 step). See df-op 4405 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof shortened by AV, 15-Jun-2022.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

TheorempreleqOLD 8813 Obsolete version of preleqALT 8811 as of 15-Jun-2022. Hypotheses 𝐴 ∈ V and 𝐵 ∈ V are not needed! (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

TheoremopthregOLD 8814 Obsolete proof of opthreg 8812 as of 15-Jun-2022. Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8788 (via the preleqOLD 8813 step). See df-op 4405 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremsuc11reg 8815 The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
(suc 𝐴 = suc 𝐵𝐴 = 𝐵)

Theoremdford2 8816* Assuming ax-reg 8788, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))

2.5.2  Axiom of Infinity equivalents

Theoreminf0 8817* 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 8834. (Contributed by NM, 15-Oct-1996.)
ω ∈ V       𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf1 8818 Variation of Axiom of Infinity (using zfinf 8835 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 8819* Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8835 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 8820* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8821* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8822* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8823* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8824* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8825* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8826* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8791. (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 8827* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8828* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8829* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 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 8830* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8831 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7417. (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 8831 Our Axiom of Infinity ax-inf 8834 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8819, 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 8836 and zfinf2 8838.) The main proof is provided by inf3lema 8820 through inf3lem7 8830, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8830. 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 8821.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8822.)
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 8824.)
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 8825.)
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 8826.)
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 8827.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8828.)
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 8829.)
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 8830.)
Q.E.D.

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

Theoreminfeq5i 8832 Half of infeq5 8833. (Contributed by Mario Carneiro, 16-Nov-2014.)
(ω ∈ V → ∃𝑥 𝑥 𝑥)

Theoreminfeq5 8833 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 8839.) 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 8834* 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 8818 and inf2 8819). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8838 and omex 8839 and are based on the (nontrivial) proof of inf3 8831. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8837. Theorem inf0 8817 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8841 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8837 requires this axiom along with Regularity ax-reg 8788 for its derivation (as theorem axinf2 8836 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8837 instead of this one. The derivation of this axiom from ax-inf2 8837 is shown by theorem axinf 8840.

Proofs should normally use the standard version ax-inf2 8837 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfinf 8835* 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 8836* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8834 and Regularity ax-reg 8788.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8837 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 8837* 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 8838 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8836 above, using our version of Infinity ax-inf 8834 and the Axiom of Regularity ax-reg 8788. We will reference ax-inf2 8837 instead of axinf2 8836 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8834 from ax-inf2 8837 is shown by theorem axinf 8840. (Contributed by NM, 3-Nov-1996.)
𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Theoremzfinf2 8838* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8837 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 8839 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 8817.

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 7356 and Fin = V (the universe of all sets) by fineqv 8465. 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 7365 through peano5 7369 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

ω ∈ V

Theoremaxinf 8840* The first version of the Axiom of Infinity ax-inf 8834 proved from the second version ax-inf2 8837. Note that we didn't use ax-reg 8788, unlike the other direction axinf2 8836. (Contributed by NM, 24-Apr-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf5 8841 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 8833). 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 8842 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
ω ∈ On

Theoremdfom3 8843* 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 8844* A simplification of elom 7348 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
(𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥𝐴𝑥))

Theoremdfom4 8845* A simplification of df-om 7346 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}

Theoremdfom5 8846 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 8847 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 8848 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 8849 A finite set is countable (weaker version of isfinite 8848). (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝐴 ∈ Fin → 𝐴 ≼ ω)

Theoremnnsdom 8850 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 8851 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 8852 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
ω ≈ suc ω

Theoreminfdifsn 8853 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 8854 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 8855* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 8506 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremnoinfep 8856* Using the Axiom of Regularity in the form zfregfr 8800, 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 8857 Extend class notation with the Cantor normal form function.
class CNF

Definitiondf-cnf 8858* 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 8891 of this function in terms of df-oi 8706. (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 8859* 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 8860* 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 8861* Lemma for cantnf 8889. The function appearing in cantnfval 8864 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
𝐹 = seq𝜔((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)), ∅)       𝐹:ω⟶On

Theoremcantnfs 8862 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 8863 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 8864* 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 8865* 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 8866* 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 8867* 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 8868* 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 8869 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 8870 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 8871 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 8872* 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 8873* 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 8874* Lemma for cantnfp1 8877. (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 8875* Lemma for cantnfp1 8877. (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 8876* Lemma for cantnfp1 8877. (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 8877* 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 8878* The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 8749). (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 8879* 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 8880* 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 8881* Lemma for cantnf 8889. (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 8882* Lemma for cantnf 8889. (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 8883* Lemma for cantnf 8889. (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 8884* Lemma for cantnf 8889. (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 8885* Lemma for cantnf 8889. 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 8886* Lemma for cantnf 8889. (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 8887* Lemma for cantnf 8889. 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 7969 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 8877 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 8888* Lemma for cantnf 8889. Complete the induction step of cantnflem3 8887. (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 8889* 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 8873, 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 8890* 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 8891* An alternate definition of df-cnf 8858 which relies on cantnf 8889. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8860 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 8892 Simplify the isomorphism of cantnf 8889 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 8893* 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 8894* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6831.) (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 8895* Lemma for cnfcom 8896. (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 8896* 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 8897* Lemma for cnfcom2 8898. (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 8898* 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 8899* Lemma for cnfcom3 8900. (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 8900* 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 8902.) (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))

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