P. 48 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opthreg 4701	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4682 (via the preleq 4700 step). See df-op 3717 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	suc11g 4702	The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	suc11 4703	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	dtruex 4704*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4326 can also be summarized as "at least two sets exist", the difference is that dtruarb 4326 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific 𝑦, we can construct a set 𝑥 which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ∃𝑥 ¬ 𝑥 = 𝑦
Theorem	dtru 4705*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4704. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	eunex 4706	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Theorem	ordsoexmid 4707	Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
⊢ E Or On    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordsuc 4708	The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
Theorem	onsucuni2 4709	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	0elsucexmid 4710*	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	nlimsucg 4711	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	ordpwsucss 4712	The collection of ordinals in the power class of an ordinal is a superset of its successor. We can think of (𝒫 𝐴 ∩ On) as another possible definition of successor, which would be equivalent to df-suc 4514 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4575) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4657). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4715). (Contributed by Jim Kingdon, 21-Jul-2019.)
⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Theorem	onnmin 4713	No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)
Theorem	ssnel 4714	Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	ordpwsucexmid 4715*	The subset in ordpwsucss 4712 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
⊢ ∀𝑥 ∈ On suc 𝑥 = (𝒫 𝑥 ∩ On)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordtri2or2exmid 4716*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ontri2orexmidim 4717*	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4716. (Contributed by Jim Kingdon, 26-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
Theorem	onintexmid 4718*	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	zfregfr 4719	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
⊢ E Fr 𝐴
Theorem	ordfr 4720	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
⊢ (Ord 𝐴 → E Fr 𝐴)
Theorem	ordwe 4721	Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
⊢ (Ord 𝐴 → E We 𝐴)
Theorem	wetriext 4722*	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	wessep 4723	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)
Theorem	reg3exmidlemwe 4724*	Lemma for reg3exmid 4725. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ E We 𝐴
Theorem	reg3exmid 4725*	If any inhabited set satisfying df-wetr 4477 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	dcextest 4726*	If it is decidable whether {𝑥 ∣ 𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
⊢ DECID {𝑥 ∣ 𝜑} ∈ V    ⇒   ⊢ DECID ¬ 𝜑
2.5.3  Transfinite induction
Theorem	tfi 4727*	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. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)
Theorem	tfis 4728*	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 4729*	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 4730*	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 4731*	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 4732*	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)
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4733*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
Theorem	zfinf2 4734*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
2.6.2  The natural numbers
Syntax	com 4735	Extend class notation to include the class of natural numbers.
class ω
Definition	df-iom 4736*	Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Note: the natural numbers ω are a subset of the ordinal numbers df-on 4511. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9288) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7540. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4737 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	dfom3 4737*	Alias for df-iom 4736. Use it instead of df-iom 4736 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	omex 4738	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
⊢ ω ∈ V
2.6.3  Peano's postulates
Theorem	peano1 4739	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
⊢ ∅ ∈ ω
Theorem	peano2 4740	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 4741	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 4742	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 4743*	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. 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 4748. (Contributed by NM, 18-Feb-2004.)
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4744*	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.)
⊢ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)    ⇒   ⊢ 𝐴 = ω
Theorem	finds 4745*	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	finds2 4746*	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 4747*	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 4748	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. 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)
2.6.5  The Natural Numbers (continued)
Theorem	nn0suc 4749*	A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Theorem	elomssom 4750	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4751. (Revised by BJ, 7-Aug-2024.)
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
Theorem	elnn 4751	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	ordom 4752	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
⊢ Ord ω
Theorem	omelon2 4753	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	omelon 4754	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ ω ∈ On
Theorem	nnon 4755	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 4756	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 4757	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	omsson 4758	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
⊢ ω ⊆ On
Theorem	limom 4759	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
⊢ Lim ω
Theorem	peano2b 4760	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 4761*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	nnsucpred 4762	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = 𝐴)
Theorem	nndceq0 4763	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)
Theorem	0elnn 4764	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	nn0eln0 4765	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	nnregexmid 4766*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4680 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6766 or nntri3or 6760), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
⊢ ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	omsinds 4767*	Strong (or "total") induction principle over ω. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ ω → 𝜒)
Theorem	nnpredcl 4768	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4709) but also holds when it is ∅ by uni0 3960. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
Theorem	nnpredlt 4769	The predecessor (see nnpredcl 4768) of a nonzero natural number is less than (see df-iord 4509) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
2.6.6  Relations
Syntax	cxp 4770	Extend the definition of a class to include the cross product.
class (𝐴 × 𝐵)
Syntax	ccnv 4771	Extend the definition of a class to include the converse of a class.
class ◡𝐴
Syntax	cdm 4772	Extend the definition of a class to include the domain of a class.
class dom 𝐴
Syntax	crn 4773	Extend the definition of a class to include the range of a class.
class ran 𝐴
Syntax	cres 4774	Extend the definition of a class to include the restriction of a class. (Read: The restriction of 𝐴 to 𝐵.)
class (𝐴 ↾ 𝐵)
Syntax	cima 4775	Extend the definition of a class to include the image of a class. (Read: The image of 𝐵 under 𝐴.)
class (𝐴 “ 𝐵)
Syntax	ccom 4776	Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.)
class (𝐴 ∘ 𝐵)
Syntax	wrel 4777	Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴
Definition	df-xp 4778*	Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of [Quine] p. 64. For example, ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}). Another example is that the set of rational numbers is defined using the Cartesian product as (ℤ × ℕ); the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
Definition	df-rel 4779	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5236 and dfrel3 5243. (Contributed by NM, 1-Aug-1994.)
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
Definition	df-cnv 4780*	Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if 𝐴 ∈ V and 𝐵 ∈ V then (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴), as proven in brcnv 4961 (see df-br 4129 and df-rel 4779 for more on relations). For example, ◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.)
⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Definition	df-co 4781*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses a slash instead of ∘, and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
Definition	df-dm 4782*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } → dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4783). For alternate definitions see dfdm2 5320, dfdm3 4965, and dfdm4 4971. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Definition	df-rn 4783	Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4782). For alternate definitions, see dfrn2 4966, dfrn3 4967, and dfrn4 5246. The notation "ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
⊢ ran 𝐴 = dom ◡𝐴
Definition	df-res 4784	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = {⟨2, 6⟩}. We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection (𝐴 ∩ (V × 𝐵)) or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
Definition	df-ima 4785	Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } /\ B = { 1 , 2 } ) -> ( F “ B ) = { 6 } . Contrast with restriction (df-res 4784) and range (df-rn 4783). For an alternate definition, see dfima2 5126. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
Theorem	xpeq1 4786	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2 4787	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	elxpi 4788*	Membership in a cross product. Uses fewer axioms than elxp 4789. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
Theorem	elxp 4789*	Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
Theorem	elxp2 4790*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	xpeq12 4791	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	xpeq1i 4792	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶)
Theorem	xpeq2i 4793	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵)
Theorem	xpeq12i 4794	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)
Theorem	xpeq1d 4795	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2d 4796	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	xpeq12d 4797	Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	sqxpeqd 4798	Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
Theorem	nfxp 4799	Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 × 𝐵)
Theorem	0nelxp 4800	The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ¬ ∅ ∈ (𝐴 × 𝐵)