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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | orduniorsuc 7801 | An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.) |
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | ||
Theorem | unon 7802 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
⊢ ∪ On = On | ||
Theorem | ordunisuc 7803 | 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 7804* | 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 7805 | 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 7806 | The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.) |
⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴) | ||
Theorem | limon 7807 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
⊢ Lim On | ||
Theorem | onuniorsuc 7808 | 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 7809 | An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ 𝐴 ⊆ On | ||
Theorem | onsuci 7810 | The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7782 and onsucb 7788. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ On | ||
Theorem | onuniorsuciOLD 7811 | Obsolete version of onuniorsuc 7808 as of 11-Jan-2025. (Contributed by NM, 13-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴) | ||
Theorem | onuninsuci 7812* | 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 7815. (Contributed by NM, 18-Feb-2004.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) | ||
Theorem | onsucssi 7813 | 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 7814 | 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 7815* | 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 7812. (Contributed by NM, 18-Feb-2004.) |
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | ||
Theorem | ordunisuc2 7816* | 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 7817* | 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 7818* | 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 7819* | 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 7820* | 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 7821 | 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 7822 | 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 7823* | 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 7824* | 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) | ||
Theorem | tfi 7825* |
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 7835 or tfinds 7832 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 7826* | A closed form of tfis 7827. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑) | ||
Theorem | tfis 7827* | 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 7828* | 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 7829* | 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 7830* | 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 7831* | 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 7832* | 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 7833* | 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 7834* | 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 7835* | 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 7836* | 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 7837* | 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)) | ||
Syntax | com 7838 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-om 7839* |
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 7840 for an alternate definition. Later, when we
assume the
Axiom of Infinity, we show ω is a set in
omex 9620, and ω can
then be defined per dfom3 9624 (the smallest inductive set) and dfom4 9626.
Note: the natural numbers ω are a subset of the ordinal numbers df-on 6357. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12195) 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 7840 | 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 7823). (Contributed by NM, 1-Nov-2004.) |
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} | ||
Theorem | elom 7841* | Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9625. (Contributed by NM, 15-May-1994.) |
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | ||
Theorem | omsson 7842 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ω ⊆ On | ||
Theorem | limomss 7843 | 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 7844 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 7845 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 7846 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | trom 7847 | 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 7848 | 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 7849 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | omon 7850 | 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 7851 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | nnlim 7852 | A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.) |
⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | ||
Theorem | omssnlim 7853 | 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 7854 | 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 7855 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 7856* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | omsucne 7857 | A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.) |
⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) | ||
Theorem | ssnlim 7858* | 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 7859* | 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 | omsindsOLD 7860* | Obsolete version of omsinds 7859 as of 16-Oct-2024. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ ω → 𝜒) | ||
Theorem | peano1 7861 | 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 7861 through peano5 7866 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 7708. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ ∅ ∈ ω | ||
Theorem | peano1OLD 7862 | Obsolete version of peano1 7861 as of 29-Nov-2024. (Contributed by NM, 15-May-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∅ ∈ ω | ||
Theorem | peano2 7863 | 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 7864 | 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.) |
⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) | ||
Theorem | peano4 7865 | 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 7866* | 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 7875. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | peano5OLD 7867* | Obsolete version of peano5 7866 as of 3-Oct-2024. (Contributed by NM, 18-Feb-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | nn0suc 7868* | A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | ||
Theorem | find 7869* | 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 | findOLD 7870* | Obsolete version of find 7869 as of 28-May-2024. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ⇒ ⊢ 𝐴 = ω | ||
Theorem | finds 7871* | 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 7872* | 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 7873* | 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 7874* | 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 7875 | 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 7835 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) | ||
Theorem | dmexg 7876 | 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 7877 | 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 7878 | The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom 𝐴 ∈ V) | ||
Theorem | fndmexd 7879 | 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 7880 | 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 7881 | 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 7882 | 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 7883 | Alternate proof of dmfex 7880: shorter but using ax-rep 5278. (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 7884 | 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 7885 | 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 7886 | 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 22690. (Contributed by NM, 1-Jan-2007.) |
⊢ ¬ I ∈ V | ||
Theorem | resiexg 7887 | The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7201). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) | ||
Theorem | imaexg 7888 | 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 7889 | 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 | exse2 7890 | Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) | ||
Theorem | xpexr 7891 | 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 7892 | 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 7893 | A condition where the converse of xpex 7723 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 7894 | 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 7895 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7896, elxp6 7991, and elxp7 7992. (Contributed by NM, 17-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | ||
Theorem | elxp5 7896 | Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7895 when the double intersection does not create class existence problems (caused by int0 4959). (Contributed by NM, 1-Aug-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∩ ∩ 𝐴, ∪ ran {𝐴}〉 ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | ||
Theorem | cnvexg 7897 | 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 7898 | 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 7899 | 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 7900 | 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) |
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