P. 96 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elnel 9501	A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
⊢ (𝐴 ∈ 𝐵 → 𝐵 ∉ 𝐴)
Theorem	en3lplem1 9502*	Lemma for en3lp 9504. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Theorem	en3lplem2 9503*	Lemma for en3lp 9504. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Theorem	en3lp 9504	No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 44876 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)
Theorem	preleqg 9505	Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9506. (Contributed by AV, 15-Jun-2022.)
⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	preleq 9506	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    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	preleqALT 9507	Alternate proof of preleq 9506, not based on preleqg 9505: 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    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthreg 9508	Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 9478 (via the preleq 9506 step). See df-op 4583 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 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	suc11reg 9509	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 𝐵 ↔ 𝐴 = 𝐵)
Theorem	dford2 9510*	Assuming ax-reg 9478, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
2.5.2  Axiom of Infinity equivalents
Theorem	inf0 9511*	Existence of ω implies our axiom of infinity ax-inf 9528. 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 9528. (Contributed by NM, 15-Oct-1996.) Revised to closed form. (Revised by BJ, 20-May-2024.)
⊢ (ω ∈ 𝑉 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
Theorem	inf1 9512	Variation of Axiom of Infinity (using zfinf 9529 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))    ⇒   ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
Theorem	inf2 9513*	Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9529 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))    ⇒   ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
Theorem	inf3lema 9514*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))
Theorem	inf3lemb 9515*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹‘∅) = ∅
Theorem	inf3lemc 9516*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹‘𝐴)))
Theorem	inf3lemd 9517*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)
Theorem	inf3lem1 9518*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))
Theorem	inf3lem2 9519*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥))
Theorem	inf3lem3 9520*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 9482. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴)))
Theorem	inf3lem4 9521*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴)))
Theorem	inf3lem5 9522*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
Theorem	inf3lem6 9523*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Theorem	inf3lem7 9524*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9525 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7889. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
Theorem	inf3 9525	Our Axiom of Infinity ax-inf 9528 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 9513, 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 9530 and zfinf2 9532.) The main proof is provided by inf3lema 9514 through inf3lem7 9524, and this final piece eliminates the auxiliary hypothesis of inf3lem7 9524. 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 9515.) F_n+1 = {y<X | y^X subset F_n} (See inf3lemc 9516.) 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 9518.) 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 9519.) 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 9520.) 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 9521.) Proof: Lemmas 1 and 3. Lemma 5. F_m proper_subset F_n, m < n. (See inf3lem5 9522.) 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 9523.) 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 9524.) Q.E.D. (Contributed by NM, 29-Oct-1996.)
⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)    ⇒   ⊢ ω ∈ V
Theorem	infeq5i 9526	Half of infeq5 9527. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
Theorem	infeq5 9527	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 9533.) 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
Axiom	ax-inf 9528*	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 9512 and inf2 9513). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9532 and omex 9533 and are based on the (nontrivial) proof of inf3 9525. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9531. Theorem inf0 9511 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9535 shows a very short way to state this axiom. The standard version of Infinity ax-inf2 9531 requires this axiom along with Regularity ax-reg 9478 for its derivation (as Theorem axinf2 9530 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9531 instead of this one. The derivation of this axiom from ax-inf2 9531 is shown by Theorem axinf 9534. Proofs should normally use the standard version ax-inf2 9531 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
Theorem	zfinf 9529*	Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
Theorem	axinf2 9530*	A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 9528 and Regularity ax-reg 9478. This theorem should not be referenced in any proof. Instead, use ax-inf2 9531 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(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Axiom	ax-inf2 9531*	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 9532 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9530 above, using our version of Infinity ax-inf 9528 and the Axiom of Regularity ax-reg 9478. We will reference ax-inf2 9531 instead of axinf2 9530 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9528 from ax-inf2 9531 is shown by Theorem axinf 9534. (Contributed by NM, 3-Nov-1996.)
⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	zfinf2 9532*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9531 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
2.6.2  Existence of omega (the set of natural numbers)
Theorem	omex 9533	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. Remark 1.21 of [Schloeder] p. 3. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 9511. 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 7808 and Fin = V (the universe of all sets) by fineqv 9151. 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 7819 through peano5 7823 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)
⊢ ω ∈ V
Theorem	axinf 9534*	The first version of the Axiom of Infinity ax-inf 9528 proved from the second version ax-inf2 9531. Note that we didn't use ax-reg 9478, unlike the other direction axinf2 9530. (Contributed by NM, 24-Apr-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
Theorem	inf5 9535	The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see Theorem infeq5 9527). 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
Theorem	omelon 9536	Omega is an ordinal number. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ ω ∈ On
Theorem	dfom3 9537*	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. Definition 1.20 of [Schloeder] p. 3. (Contributed by NM, 6-Aug-1994.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	elom3 9538*	A simplification of elom 7799 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))
Theorem	dfom4 9539*	A simplification of df-om 7797 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom5 9540	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). Theorem 1.23 of [Schloeder] p. 4. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}
Theorem	oancom 9541	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)
Theorem	isfinite 9542	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)
Theorem	fict 9543	A finite set is countable (weaker version of isfinite 9542). (Contributed by Thierry Arnoux, 27-Mar-2018.)
⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω)
Theorem	nnsdom 9544	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)
Theorem	omenps 9545	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 ∖ {∅})
Theorem	omensuc 9546	The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
⊢ ω ≈ suc ω
Theorem	infdifsn 9547	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)
Theorem	infdiffi 9548	Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)
Theorem	unbnn3 9549*	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 9180 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)
Theorem	noinfep 9550*	Using the Axiom of Regularity in the form zfregfr 9494, 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) ∉ (𝐹‘𝑥)
2.6.3  Cantor normal form
Syntax	ccnf 9551	Extend class notation with the Cantor normal form function.
class CNF
Definition	df-cnf 9552*	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 𝑥 ↑o 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 9585 of this function in terms of df-oi 9396. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnffval 9553*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnfdm 9554*	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 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Theorem	cantnfvalf 9555*	Lemma for cantnf 9583. The function appearing in cantnfval 9558 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)    ⇒   ⊢ 𝐹:ω⟶On
Theorem	cantnfs 9556	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
Theorem	cantnfcl 9557	Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
Theorem	cantnfval 9558*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Theorem	cantnfval2 9559*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
Theorem	cantnfsuc 9560*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
Theorem	cantnfle 9561*	A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) 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 𝐶 ∈ 𝐵 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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
Theorem	cantnflt 9562*	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 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnflt2 9563	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnff 9564	The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
Theorem	cantnf0 9565	The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
Theorem	cantnfrescl 9566*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
Theorem	cantnfres 9567*	The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    &   ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))
Theorem	cantnfp1lem1 9568*	Lemma for cantnfp1 9571. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	cantnfp1lem2 9569*	Lemma for cantnfp1 9571. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    ⇒   ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂)
Theorem	cantnfp1lem3 9570*	Lemma for cantnfp1 9571. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅)    &   ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
Theorem	cantnfp1 9571*	If 𝐹 is created by adding a single term (𝐹‘𝑋) = 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 ((𝐴 ↑o 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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
Theorem	oemapso 9572*	The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9439). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → 𝑇 Or 𝑆)
Theorem	oemapval 9573*	Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
Theorem	oemapvali 9574*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
Theorem	cantnflem1a 9575*	Lemma for cantnf 9583. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
Theorem	cantnflem1b 9576*	Lemma for cantnf 9583. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))
Theorem	cantnflem1c 9577*	Lemma for cantnf 9583. (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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
Theorem	cantnflem1d 9578*	Lemma for cantnf 9583. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))
Theorem	cantnflem1 9579*	Lemma for cantnf 9583. 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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺))
Theorem	cantnflem2 9580*	Lemma for cantnf 9583. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Theorem	cantnflem3 9581*	Lemma for cantnf 9583. 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 8518 to factor 𝐶 into the form ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑o 𝑋) ≤ (𝐴 ↑o 𝑋) ·o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑o 𝑋) ·o 𝑌 using cantnfp1 9571 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnflem4 9582*	Lemma for cantnf 9583. Complete the induction step of cantnflem3 9581. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnf 9583*	The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑o 𝑓(𝑎1)) ∘ 𝑎1) +o ((𝐴 ↑o 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 (𝐴 ↑o 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 9567, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)))
Theorem	oemapwe 9584*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)))
Theorem	cantnffval2 9585*	An alternate definition of df-cnf 9552 which relies on cantnf 9583. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9554 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))
Theorem	cantnff1o 9586	Simplify the isomorphism of cantnf 9583 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵))
Theorem	wemapwe 9587*	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⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑆 We 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ 𝐹 = OrdIso(𝑅, 𝐴)    &   ⊢ 𝐺 = OrdIso(𝑆, 𝐵)    &   ⊢ 𝑍 = (𝐺‘∅)    ⇒   ⊢ (𝜑 → 𝑇 We 𝑈)
Theorem	oef1o 9588*	A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7236.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))    &   ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
Theorem	cnfcomlem 9589*	Lemma for cnfcom 9590. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼)))    &   ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom 9590*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom2lem 9591*	Lemma for cnfcom2 9592. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
Theorem	cnfcom2 9592*	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)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
Theorem	cnfcom3lem 9593*	Lemma for cnfcom3 9594. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
Theorem	cnfcom3 9594*	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 9596.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    ⇒   ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊))
Theorem	cnfcom3clem 9595*	Lemma for cnfcom3c 9596. (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 (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    &   ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)    ⇒   ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
Theorem	cnfcom3c 9596*	Wrap the construction of cnfcom3 9594 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 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
2.6.4  Transitive closure of a relation
Syntax	cttrcl 9597	Declare the syntax for the transitive closure of a class.
class t++𝑅
Definition	df-ttrcl 9598*	Define the transitive closure of a class. This is the smallest relation containing 𝑅 (or more precisely, the relation (𝑅 ↾ V) induced by 𝑅) and having the transitive property. Definition from [Levy] p. 59, who denotes it as 𝑅∗ and calls it the "ancestral" of 𝑅. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
Theorem	ttrcleq 9599	Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)
Theorem	nfttrcld 9600	Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝜑 → Ⅎ𝑥𝑅)    ⇒   ⊢ (𝜑 → Ⅎ𝑥t++𝑅)