P. 100 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hfsn 9901	The singleton of a hereditarily finite set is a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ (𝐴 ∈ HF → {𝐴} ∈ HF )
Theorem	hfsnOLD 9902	Obsolete version of hfsn 9901 as of 17-Sep-2026. (Contributed by Scott Fenton, 15-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ HF → {𝐴} ∈ HF )
Theorem	hfadj 9903	Adjoining one hereditarily finite set to a hereditarily finite set results in a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ {𝐵}) ∈ HF )
Theorem	elhf3OLD 9904	Obsolete version of elhf3 9894 as of 17-Sep-2026. (Contributed by Eric Schmidt, 8-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ 𝐴 ⊆ HF ))
Theorem	hfuni 9905	The union of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ (𝐴 ∈ HF → ∪ 𝐴 ∈ HF )
Theorem	hfuniOLD 9906	Obsolete version of hfuni 9905 as of 17-Sep-2026. (Contributed by Scott Fenton, 16-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ HF → ∪ 𝐴 ∈ HF )
Theorem	hfpw 9907	The power class of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ (𝐴 ∈ HF → 𝒫 𝐴 ∈ HF )
Theorem	hfpwOLD 9908	Obsolete version of hfpw 9907 as of 17-Sep-2026. (Contributed by Scott Fenton, 16-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ HF → 𝒫 𝐴 ∈ HF )
2.6.11  Scott's trick; collection principle; Hilbert's epsilon
Syntax	cscott 9909	Extend class notation with the Scott's trick operation.
class Scott 𝐴
Definition	df-scott 9910*	Define an operation for Scott's trick, expressed as in Equation 9.3 of [Jech] p. 72. Scott's trick collects all sets that have a certain property and are of the smallest possible rank. The resulting collection is guaranteed to be a set (see scottex 9914). Under this definition, the property in question is represented by membership in a class. See scottab 9921 for the case where a wff (meta)variable is used to represent the property. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
Theorem	scotteqd 9911	Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)
Theorem	scotteq 9912	Closed form of scotteqd 9911. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵)
Theorem	nfscott 9913	Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Scott 𝐴
Theorem	scottex 9914	Scott's trick produces a set. (Contributed by NM, 13-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 18-Jul-2026.)
⊢ Scott 𝐴 ∈ V
Theorem	scottexOLD 9915*	Obsolete version of scottex 9914 as of 18-Jul-2026. (Contributed by NM, 13-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Theorem	scottss 9916	Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Scott 𝐴 ⊆ 𝐴
Theorem	scott0 9917	Applying Scott's trick to the empty set leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ Scott ∅ = ∅
Theorem	scott0b 9918	Applying Scott's trick yields the empty set iff it was applied to the empty set. (Contributed by NM, 15-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
Theorem	scott0OLD 9919*	Obsolete version of scott0b 9918 as of 19-Jul-2026. (Contributed by NM, 15-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Theorem	scottabf 9920*	Value of the Scott operation at a class abstraction. Variant of scottab 9921 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Theorem	scottab 9921*	Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Theorem	scottabes 9922*	Value of the Scott operation at a class abstraction. Variant of scottab 9921 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Theorem	elscottab 9923*	An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓)
Theorem	scottexsOLD 9924*	Obsolete theorem as of 27-Jul-2026. Use scottex 9914 with a class abstraction instead. (Contributed by NM, 13-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Theorem	scott0bs 9925	Theorem scheme version of scott0b 9918. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 22-Jul-2026.)
⊢ (∃𝑥𝜑 ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
Theorem	scott0bsOLD 9926*	Obsolete version of scott0bs 9925 as of 19-Jul-2026. (Contributed by NM, 13-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
Theorem	scottex2OLD 9927	Obsolete version of scottex 9914 as of 18-Jul-2026. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Scott 𝐴 ∈ V
Theorem	scotteld 9928*	The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
Theorem	scottelrankd 9929	Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴)    ⇒   ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))
Theorem	scottrankd 9930	Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)    ⇒   ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))
Theorem	cplem1 9931	Lemma for the Collection Principle cp 9935. (Contributed by NM, 17-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
⊢ 𝐶 = ∪ 𝑥 ∈ 𝐴 Scott 𝐵    ⇒   ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐶) ≠ ∅)
Theorem	cplem1OLD 9932*	Obsolete version of cplem1 9931 as of 19-Jul-2026. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)}    &   ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶    ⇒   ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅)
Theorem	cplem2 9933*	Lemma for the Collection Principle cp 9935. (Contributed by NM, 17-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)
Theorem	cplem2OLD 9934*	Obsolete version of cplem2 9933 as of 19-Jul-2026. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)
Theorem	cp 9935*	Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick df-scott 9910 to collapse a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd 9936*	A very strong generalization of the Axiom of Replacement (compare zfrep6 5242), derived from the Collection Principle cp 9935. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd2 9937*	A variant of the Boundedness Axiom bnd 9936 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))
Theorem	kardex 9938	The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
⊢ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ∈ V
Theorem	kardexOLD 9939*	Obsolete version of kardex 9938 as of 19-Jul-2026. (Contributed by NM, 14-Dec-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Theorem	karden 9940*	If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10616). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9938 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}    &   ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}    ⇒   ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)
Theorem	kardenOLD 9941*	Obsolete version of karden 9940 as of 19-Jul-2026. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    ⇒   ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)
Theorem	htalem 9942*	Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom", described on that page, with the additional 𝑅 We 𝐴 antecedent. The element 𝐵 is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)
Theorem	hta 9943*	A ZFC emulation of Hilbert's transfinite axiom. The set 𝐵 has the properties of Hilbert's epsilon, except that it also depends on a well-ordering 𝑅. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See https://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and https://us.metamath.org/downloads/megillaward2005he.pdf. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires 𝑅 We 𝐴 as an antecedent. Class 𝐴 collects the sets of the least rank for which 𝜑(𝑥) is true. Class 𝐵, which emulates Hilbert's epsilon, is the minimum element in a well-ordering 𝑅 on 𝐴. If a well-ordering 𝑅 on 𝐴 can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace 𝑅 with a dummy setvar variable, say 𝑤, and attach 𝑤 We 𝐴 as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, 𝐵 (which will have 𝑤 as a free variable) will no longer be present, and we can eliminate 𝑤 We 𝐴 by applying exlimiv 1963 and weth 10554, using scottex 9914 to establish the existence of 𝐴. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9942. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) Use the Scott operation. (Revised by BTernaryTau, 22-Jul-2026.)
⊢ 𝐴 = Scott {𝑥 ∣ 𝜑}    &   ⊢ 𝐵 = (℩𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑧𝑅𝑦)    ⇒   ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))
Theorem	htaOLD 9944*	Obsolete version of hta 9943 as of 22-Jul-2026. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧)    ⇒   ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))
2.6.12  Set Recursion
2.6.12.1  Basic Properties of Set Recursion Symbols in this section: All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 9946. The class 𝑌 is explained in the comment of setrec1lem1 9947. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.
Syntax	csetrecs 9945	Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
Definition	df-setrecs 9946*	Define a class setrecs(𝐹) by transfinite recursion, where (𝐹‘𝑥) is the set of new elements to add to the class given the set 𝑥 of elements in the class so far. We do not need a base case, because we can start with the empty set, which is vacuously a subset of setrecs(𝐹). The goal of this definition is to construct a class fulfilling Theorems setrec1 9953 and setrec2v 9959, which give a more intuitive idea of the meaning of setrecs. Unlike wrecs, setrecs is well-defined for any 𝐹 and meaningful for any function 𝐹. For example, see Theorem onsetrec 50745 for how the class On is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤 ⊆ 𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2o} would fail the condition since 2o is not an element of the successor of ∅ or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is ∅, and suc ∅ does not have 1o as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with ∅ and repeatedly applying the successor function. A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ∪ ran recs((𝑔 ∈ V ↦ (𝐹‘∪ ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 9953 and setrec2v 9959 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant". Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within Metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, Metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the Metamath framework. The difficulty in writing a definition in Metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 9953 and setrec2v 9959), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 8402, this is not actually a requirement of the Metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction. If setrecs(𝐹) = ∪ 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹. One necessary condition is that each element of a given 𝑥 ∈ 𝐴 must be generated by 𝐹 when applied to a previous element 𝑦 ∈ 𝐴. In symbols, ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐴(𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐹‘𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦 ∈ 𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have ∪ 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴. The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹‘𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥∀𝑧 ∈ 𝑥¬ (𝐹‘𝑧) ⊆ 𝑦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝑦 ∈ 𝑧)} In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. ∪ {𝑦 ∣ ∀𝑧 (𝑦 ∈ 𝑧 → ∃𝑤(𝑤 ⊆ 𝑦 ∧ (𝐹‘𝑤) ∈ 𝑧 ∧ ¬ 𝑤 ∈ 𝑧))} These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦 ∈ 𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of ∈ 𝑧 with ⊆ 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤 ∈ 𝑧, so the definition requires the additional constraint (𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) in order to work. This issue can also be avoided by replacing ∈ 𝑧 with ⊆ 𝑧. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(𝑦 ∩ 𝑧 ≠ ∅ → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∩ 𝑧 ≠ ∅))} In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹‘𝑤) for 𝑤 ⊆ (𝑦 ∩ 𝑧) even if those elements of (𝐹‘𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹‘𝑤) in the definition by ((𝐹‘𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦 ⊆ 𝑧 would be false. We could then simply add all elements of (𝐹‘𝑤) outside of 𝑦 for any 𝑤 ⊆ 𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹‘𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦 ∩ 𝑧), and in particular ((𝐹‘𝑤) ∩ 𝑦) ⊆ 𝑧. The foundedness version is starting to look more like ax-reg 9570! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣 ∈ 𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ 𝑣 ∈ (𝐹‘𝑤)))} Now, if we replace (𝐹‘𝑤) by ((𝐹‘𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣 ∈ 𝑦 since 𝑣 ∈ 𝑧 and 𝑧 ⊆ 𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5604, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧. Version 4 - Foundedness (6-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤∀𝑢 ∈ 𝑧(𝑤 ⊆ 𝑦 ∧ ¬ 𝑢 ∈ 𝑤 ∧ 𝑣 ∈ (𝐹‘𝑤)) This is so close to df-fr 5604; the only change needed is to switch ∃𝑤 with ∀𝑢 ∈ 𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9570. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of ∪ 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦 ⊆ 𝑧 rather than only 𝑦 ∈ 𝑧. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(∃𝑎(𝑎 ⊆ 𝑦 ∧ 𝑎 ∈ 𝑧) → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. ) Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of ∪ 𝑧. I think this has the same problem. We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} Now this looks more like df-fr 5604! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹‘𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5604. This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧. Version 1.3 (31-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝑤 ∈ 𝑧 ∧ (𝐹‘𝑤) ∈ 𝑧))) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝑤 ∈ 𝑧 ∨ (𝐹‘𝑤) ∈ 𝑧)))} 𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work. Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹‘𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹‘𝑤) corresponds to the variable 𝑦 in df-fr 5604. Thus, in our case, df-fr 5604 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧(𝐹‘𝑤) ∈ (𝐹‘𝑣))) This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑣 ∈ 𝑧¬ 𝑣𝑅(𝐹‘𝑤) so we can replace one with the other. Negate both sides: 𝑤 ∈ 𝑧 ↔ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤) If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.)
⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
Theorem	setrec1lem1 9947*	Lemma for setrec1 9953. This is a utility theorem showing the equivalence of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof uses elabg 3630 and equivalence theorems. Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
Theorem	setrec1lem2 9948*	Lemma for setrec1 9953. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3920, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌)
Theorem	bnd2d 9949*	Deduction form of bnd2 9937. (Contributed by Emmett Weisz, 19-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓))
Theorem	setrec1lem3 9950*	Lemma for setrec1 9953. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 9948. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴 ∈ 𝑌. I don't know if proving this fact directly using setrec1lem1 9947 would be any easier than the current proof using setrec1lem2 9948, and it would only slightly simplify the proof of setrec1 9953. Other than the use of bnd2d 9949, this is a purely technical theorem for rearranging notation from that of setrec1lem2 9948 to that of setrec1 9953. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌))    ⇒   ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
Theorem	spcdvw 9951*	A version of spcdv 3549 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	setrec1lem4 9952*	Lemma for setrec1 9953. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴). In the proof of setrec1 9953, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis Ⅎ𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1868, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 9948 could similarly be made easier to read by adding the hypothesis Ⅎ𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.)
⊢ Ⅎ𝑧𝜑    &   ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌)
Theorem	setrec1 9953	This is the first of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is closed under 𝐹. This effectively sets the actual value of setrecs(𝐹) as a lower bound for setrecs(𝐹), as it implies that any set generated by successive applications of 𝐹 is a member of 𝐵. This theorem "gets off the ground" because we can start by letting 𝐴 = ∅, and the hypotheses of the theorem will hold trivially. Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon). Proof summary: Assume that 𝐴 ⊆ 𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 9950, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 9950.) Therefore, by setrec1lem4 9952, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4893, it is a subset of the union of all sets recursively generated by 𝐹. See df-setrecs 9946 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.)
⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵)
Theorem	setrec2fun 9954*	This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, Theorems setrec1 9953 and setrec2v 9959 say that setrecs(𝐹) is the minimal class closed under 𝐹. We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7853) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)
⊢ Ⅎ𝑎𝐹    &   ⊢ 𝐵 = setrecs(𝐹)    &   ⊢ Fun 𝐹    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	setrec2lem1 9955*	Lemma for setrec2 9958. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)
Theorem	dffun3f 9956*	Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑧𝐴    ⇒   ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
Theorem	setrec2lem2 9957*	Lemma for setrec2 9958. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Theorem	setrec2 9958*	This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, Theorems setrec1 9953 and setrec2v 9959 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹. We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7853) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.)
⊢ Ⅎ𝑎𝐹    &   ⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	setrec2v 9959*	Version of setrec2 9958 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
2.6.13  Disjoint union
Syntax	cdju 9960	Extend class notation to include disjoint union of two classes.
class (𝐴 ⊔ 𝐵)
Syntax	cinl 9961	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 9962	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-dju 9963	Disjoint union of two classes. This is a way of creating a set which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
Definition	df-inl 9964	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
Definition	df-inr 9965	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
Theorem	djueq12 9966	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))
Theorem	djueq1 9967	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))
Theorem	djueq2 9968	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
Theorem	nfdju 9969	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)
Theorem	djuex 9970	The disjoint union of sets is a set. For a shorter proof using djuss 9982 see djuexALT 9984. (Contributed by AV, 28-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djuexb 9971	The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djulcl 9972	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurcl 9973	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djulf1o 9974	The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inl:V–1-1-onto→({∅} × V)
Theorem	djurf1o 9975	The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inr:V–1-1-onto→({1o} × V)
Theorem	inlresf 9976	The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
Theorem	inlresf1 9977	The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)
Theorem	inrresf 9978	The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
Theorem	inrresf1 9979	The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵)
Theorem	djuin 9980	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Theorem	djur 9981*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
Theorem	djuss 9982	A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	djuunxp 9983	The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.)
⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	djuexALT 9984	Alternate proof of djuex 9970, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	eldju1st 9985	The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.)
⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
Theorem	eldju2ndl 9986	The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴)
Theorem	eldju2ndr 9987	The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵)
Theorem	djuun 9988	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
Theorem	1stinl 9989	The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Theorem	2ndinl 9990	The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Theorem	1stinr 9991	The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)
Theorem	2ndinr 9992	The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Theorem	updjudhf 9993*	The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
Theorem	updjudhcoinlf 9994*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Theorem	updjudhcoinrg 9995*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
Theorem	updjud 9996*	Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9964 and df-inr 9965, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9965 (25-Aug-2023). This is a special case of Example 1 of coproducts in Section 10.67 of [Adamek] p. 185. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
2.6.14  Cardinal numbers
Syntax	ccrd 9997	Extend class definition to include the cardinal size function.
class card
Syntax	cale 9998	Extend class definition to include the aleph function.
class ℵ
Syntax	ccf 9999	Extend class definition to include the cofinality function.
class cf
Syntax	wacn 10000	The axiom of choice for limited-length sequences.
class AC 𝐴