P. 79 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ordunisuc 7801	An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
Theorem	orduniss2 7802*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)
Theorem	onsucuni2 7803	A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)
Theorem	0elsuc 7804	The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)
⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴)
Theorem	limon 7805	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
⊢ Lim On
Theorem	onuniorsuc 7806	An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.) Put in closed form. (Revised by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
Theorem	onssi 7807	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ 𝐴 ⊆ On
Theorem	onsuci 7808	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7782 and onsucb 7786. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ On
Theorem	onuninsuci 7809*	An ordinal is equal to its union if and only if it is not the successor of an ordinal. A closed-form generalization of this result is orduninsuc 7812. (Contributed by NM, 18-Feb-2004.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Theorem	onsucssi 7810	A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)
Theorem	nlimsucg 7811	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)
Theorem	orduninsuc 7812*	An ordinal class is equal to its union if and only if it is not the successor of an ordinal. Closed-form generalization of onuninsuci 7809. (Contributed by NM, 18-Feb-2004.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
Theorem	ordunisuc2 7813*	An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
Theorem	ordzsl 7814*	An ordinal is zero, a successor ordinal, or a limit ordinal. Remark 1.12 of [Schloeder] p. 2. (Contributed by NM, 1-Oct-2003.)
⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Theorem	onzsl 7815*	An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Theorem	dflim3 7816*	An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
Theorem	dflim4 7817*	An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
Theorem	limsuc 7818	The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
Theorem	limsssuc 7819	A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))
Theorem	nlimon 7820*	Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)
⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}
Theorem	limuni3 7821*	The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)
2.4.3  Transfinite induction
Theorem	tfi 7822*	The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if 𝐴 is a class of ordinal numbers with the property that every ordinal number included in 𝐴 also belongs to 𝐴, then every ordinal number is in 𝐴. See Theorem tfindes 7832 or tfinds 7829 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)
Theorem	tfisg 7823*	A closed form of tfis 7824. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)
Theorem	tfis 7824*	Transfinite Induction Schema. If all ordinal numbers less than a given number 𝑥 have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2f 7825*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2 7826*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis3 7827*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜒)
Theorem	tfisi 7828*	A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ On)    &   ⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	tfinds 7829*	Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. Theorem 1.19 of [Schloeder] p. 3. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ On → (𝜒 → 𝜃))    &   ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜏)
Theorem	tfindsg 7830*	Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal 𝐵 instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))    &   ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → 𝜏)
Theorem	tfindsg2 7831*	Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal suc 𝐵 instead of zero. (Contributed by NM, 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = suc 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))    &   ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)
Theorem	tfindes 7832*	Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
⊢ [∅ / 𝑥]𝜑    &   ⊢ (𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑))    &   ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfinds2 7833*	Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff 𝜏 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜏 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜏 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))
Theorem	tfinds3 7834*	Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝜂 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))
2.4.4  The natural numbers (i.e., finite ordinals)
Syntax	com 7835	Extend class notation to include the class of natural numbers.
class ω
Definition	df-om 7836*	Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e., all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 7837 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9588, and ω can then be defined per dfom3 9592 (the smallest inductive set) and dfom4 9594. Note: the natural numbers ω are a subset of the ordinal numbers df-on 6339. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12201) with analogous properties and operations, but they will be different sets. (Contributed by NM, 15-May-1994.)
⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom2 7837	An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the restricted class abstraction of non-limit ordinal numbers (see nlimon 7820). (Contributed by NM, 1-Nov-2004.)
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Theorem	elom 7838*	Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9593. (Contributed by NM, 15-May-1994.)
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
Theorem	omsson 7839	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ω ⊆ On
Theorem	limomss 7840	The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
⊢ (Lim 𝐴 → ω ⊆ 𝐴)
Theorem	nnon 7841	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 7842	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 7843	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	trom 7844	The class of finite ordinals ω is a transitive class. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Tr ω
Theorem	ordom 7845	The class of finite ordinals ω is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Ord ω
Theorem	elnn 7846	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	omon 7847	The class of natural numbers ω is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)
⊢ (ω ∈ On ∨ ω = On)
Theorem	omelon2 7848	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	nnlim 7849	A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
⊢ (𝐴 ∈ ω → ¬ Lim 𝐴)
Theorem	omssnlim 7850	The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}
Theorem	limom 7851	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Theorem 1.23 of [Schloeder] p. 4. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ Lim ω
Theorem	peano2b 7852	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 7853*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	omsucne 7854	A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
Theorem	ssnlim 7855*	An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)
⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
Theorem	omsinds 7856*	Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ ω → 𝜒)
Theorem	omun 7857	The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)
2.4.5  Peano's postulates
Theorem	peano1 7858	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7858 through peano5 7863 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) Avoid ax-un 7707. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ∅ ∈ ω
Theorem	peano2 7859	The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano3 7860	The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) Avoid ax-nul 5250. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Theorem	peano3OLD 7861	Obsolete version of peano3 7860 as of 10-Jun-2026. (Contributed by NM, 3-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Theorem	peano4 7862	Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	peano5 7863*	The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 7870. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2169, ax-12 2206. (Revised by GG, 3-Oct-2024.)
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	nn0suc 7864*	A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2.4.6  Finite induction (for finite ordinals)
Theorem	find 7865*	The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that 𝐴 is a set of natural numbers, zero belongs to 𝐴, and given any member of 𝐴 the member's successor also belongs to 𝐴. The conclusion is that every natural number is in 𝐴. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Wolf Lammen, 28-May-2024.)
⊢ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)    ⇒   ⊢ 𝐴 = ω
Theorem	finds 7866*	Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ω → 𝜏)
Theorem	findsg 7867*	Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. The basis of this version is an arbitrary natural number 𝐵 instead of zero. (Contributed by NM, 16-Sep-1995.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ ω → 𝜓)    &   ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))    ⇒   ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → 𝜏)
Theorem	finds2 7868*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))
Theorem	finds1 7869*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
Theorem	findes 7870	Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. See tfindes 7832 for the transfinite version. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
⊢ [∅ / 𝑥]𝜑    &   ⊢ (𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
2.4.7  Relations and functions (cont.)
Theorem	dmexg 7871	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
Theorem	rnexg 7872	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
Theorem	dmexd 7873	The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 ∈ V)
Theorem	fndmexd 7874	If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	dmfex 7875	If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
Theorem	fndmexb 7876	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
Theorem	fdmexb 7877	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
Theorem	dmfexALT 7878	Alternate proof of dmfex 7875: shorter but using ax-rep 5221. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 23-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
Theorem	dmex 7879	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
⊢ 𝐴 ∈ V    ⇒   ⊢ dom 𝐴 ∈ V
Theorem	rnex 7880	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ran 𝐴 ∈ V
Theorem	iprc 7881	The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 23290. (Contributed by NM, 1-Jan-2007.)
⊢ ¬ I ∈ V
Theorem	resiexg 7882	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7188). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
Theorem	imaexg 7883	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
Theorem	imaex 7884	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 “ 𝐵) ∈ V
Theorem	rnexd 7885	The range of a set is a set. Deduction version of rnexd 7885. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran 𝐴 ∈ V)
Theorem	imaexd 7886	The image of a set is a set. Deduction version of imaexg 7883. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)
Theorem	exse2 7887	Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	xpexr 7888	If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))
Theorem	xpexr2 7889	If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)
⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	xpexcnv 7890	A condition where the converse of xpex 7725 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
Theorem	soex 7891	If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)
⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) → 𝐴 ∈ V)
Theorem	elxp4 7892	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7893, elxp6 7993, and elxp7 7994. (Contributed by NM, 17-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	elxp5 7893	Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7892 when the double intersection does not create class existence problems (caused by int0 4914). (Contributed by NM, 1-Aug-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	cnvexg 7894	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
Theorem	cnvex 7895	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ◡𝐴 ∈ V
Theorem	relcnvexb 7896	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))
Theorem	f1oexrnex 7897	If the range of a 1-1 onto function is a set, the function itself is a set. (Contributed by AV, 2-Jun-2019.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ V)
Theorem	f1oexbi 7898*	There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
Theorem	coexg 7899	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 7900	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V