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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ordelsuc 7801 | A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | ||
Theorem | onsucmin 7802* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥}) | ||
Theorem | ordsucelsuc 7803 | Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) | ||
Theorem | ordsucsssuc 7804 | The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) | ||
Theorem | ordsucuniel 7805 | Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.) |
⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) | ||
Theorem | ordsucun 7806 | The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵)) | ||
Theorem | ordunpr 7807 | The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) | ||
Theorem | ordunel 7808 | The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 ∪ 𝐶) ∈ 𝐴) | ||
Theorem | onsucuni 7809 | A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.) |
⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | ||
Theorem | ordsucuni 7810 | An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.) |
⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴) | ||
Theorem | orduniorsuc 7811 | 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 7812 | 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 7813 | 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 7814* | 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 7815 | 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 7816 | The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.) |
⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴) | ||
Theorem | limon 7817 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
⊢ Lim On | ||
Theorem | onuniorsuc 7818 | 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 7819 | An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ 𝐴 ⊆ On | ||
Theorem | onsuci 7820 | The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7792 and onsucb 7798. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ On | ||
Theorem | onuniorsuciOLD 7821 | Obsolete version of onuniorsuc 7818 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 7822* | 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 7825. (Contributed by NM, 18-Feb-2004.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) | ||
Theorem | onsucssi 7823 | 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 7824 | 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 7825* | 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 7822. (Contributed by NM, 18-Feb-2004.) |
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | ||
Theorem | ordunisuc2 7826* | 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 7827* | 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 7828* | 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 7829* | 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 7830* | 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 7831 | 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 7832 | 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 7833* | 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 7834* | 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 7835* |
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 7845 or tfinds 7842 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 7836* | A closed form of tfis 7837. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑) | ||
Theorem | tfis 7837* | 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 7838* | 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 7839* | 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 7840* | 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 7841* | 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 7842* | 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 7843* | 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 7844* | 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 7845* | 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 7846* | 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 7847* | 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 7848 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-om 7849* |
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 7850 for an alternate definition. Later, when we
assume the
Axiom of Infinity, we show ω is a set in
omex 9634, and ω can
then be defined per dfom3 9638 (the smallest inductive set) and dfom4 9640.
Note: the natural numbers ω are a subset of the ordinal numbers df-on 6358. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12210) 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 7850 | 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 7833). (Contributed by NM, 1-Nov-2004.) |
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} | ||
Theorem | elom 7851* | Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9639. (Contributed by NM, 15-May-1994.) |
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | ||
Theorem | omsson 7852 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ω ⊆ On | ||
Theorem | limomss 7853 | 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 7854 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 7855 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 7856 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | trom 7857 | 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 7858 | 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 7859 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | omon 7860 | 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 7861 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | nnlim 7862 | A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.) |
⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | ||
Theorem | omssnlim 7863 | 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 7864 | 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 7865 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 7866* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | omsucne 7867 | A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.) |
⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) | ||
Theorem | ssnlim 7868* | 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 7869* | 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 7870* | Obsolete version of omsinds 7869 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 | omun 7871 | The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω) | ||
Theorem | peano1 7872 | 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 7872 through peano5 7877 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 7718. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ ∅ ∈ ω | ||
Theorem | peano1OLD 7873 | Obsolete version of peano1 7872 as of 29-Nov-2024. (Contributed by NM, 15-May-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∅ ∈ ω | ||
Theorem | peano2 7874 | 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 7875 | 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 7876 | 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 7877* | 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 7886. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2129, ax-12 2163. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | peano5OLD 7878* | Obsolete version of peano5 7877 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 7879* | 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 7880* | 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 7881* | Obsolete version of find 7880 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 7882* | 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 7883* | 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 7884* | 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 7885* | 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 7886 | 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 7845 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 7887 | 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 7888 | 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 7889 | The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom 𝐴 ∈ V) | ||
Theorem | fndmexd 7890 | 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 7891 | 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 7892 | 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 7893 | 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 7894 | Alternate proof of dmfex 7891: shorter but using ax-rep 5275. (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 7895 | 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 7896 | 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 7897 | 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 23083. (Contributed by NM, 1-Jan-2007.) |
⊢ ¬ I ∈ V | ||
Theorem | resiexg 7898 | The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7208). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) | ||
Theorem | imaexg 7899 | 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 7900 | 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 |
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