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Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreuhypd 4401* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremreuhyp 4402* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremuniexb 4403 The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ V ↔ 𝐴 ∈ V)

Theorempwexb 4404 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Theoremelpwpwel 4405 A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
(𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)

Theoremuniv 4406 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
V = V

Theoremeldifpw 4407 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
𝐶 ∈ V       ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))

Theoremop1stb 4408 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴

Theoremop1stbg 4409 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)

Theoremiunpw 4410* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝐴 ∈ V       (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)

2.4.2  Ordinals (continued)

Theoremordon 4411 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Ord On

Theoremssorduni 4412 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(𝐴 ⊆ On → Ord 𝐴)

Theoremssonuni 4413 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
(𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))

Theoremssonunii 4414 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 ∈ On)

Theoremonun2 4415 The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)

Theoremonun2i 4416 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵) ∈ On

Theoremordsson 4417 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)
(Ord 𝐴𝐴 ⊆ On)

Theoremonss 4418 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)

Theoremonuni 4419 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)

Theoremorduni 4420 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)

Theorembm2.5ii 4421* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})

Theoremsucexb 4422 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)

Theoremsucexg 4423 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theoremsucex 4424 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V

Theoremordsucim 4425 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
(Ord 𝐴 → Ord suc 𝐴)

Theoremsuceloni 4426 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ On → suc 𝐴 ∈ On)

Theoremordsucg 4427 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
(𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))

Theoremsucelon 4428 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Theoremordsucss 4429 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
(Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))

Theoremordelsuc 4430 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.)
((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))

Theoremonsucssi 4431 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       (𝐴𝐵 ↔ suc 𝐴𝐵)

Theoremonsucmin 4432* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
(𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})

Theoremonsucelsucr 4433 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4454. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6396. (Contributed by Jim Kingdon, 17-Jul-2019.)
(𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))

Theoremonsucsssucr 4434 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4451. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))

Theoremsucunielr 4435 Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4455. (Contributed by Jim Kingdon, 2-Aug-2019.)
(suc 𝐴𝐵𝐴 𝐵)

Theoremunon 4436 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
On = On

Theoremonuniss2 4437* The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
(𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)

Theoremlimon 4438 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Lim On

Theoremordunisuc2r 4439* An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
(Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))

Theoremonssi 4440 An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       𝐴 ⊆ On

Theoremonsuci 4441 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On       suc 𝐴 ∈ On

Theoremonintonm 4442* The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)

Theoremonintrab2im 4443 An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)

Theoremordtriexmidlem 4444 Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4446 or weak linearity in ordsoexmid 4486) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
{𝑥 ∈ {∅} ∣ 𝜑} ∈ On

Theoremordtriexmidlem2 4445* Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4446 or weak linearity in ordsoexmid 4486) with a proposition 𝜑. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Theoremordtriexmid 4446* Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

Theoremordtri2orexmid 4447* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

Theorem2ordpr 4448 Version of 2on 6331 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Ord {∅, {∅}}

Theoremontr2exmid 4449* An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)       (𝜑 ∨ ¬ 𝜑)

Theoremordtri2or2exmidlem 4450* A set which is 2o if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
{𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

Theoremonsucsssucexmid 4451* The converse of onsucsssucr 4434 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦)       (𝜑 ∨ ¬ 𝜑)

Theoremonsucelsucexmidlem1 4452* Lemma for onsucelsucexmid 4454. (Contributed by Jim Kingdon, 2-Aug-2019.)
∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}

Theoremonsucelsucexmidlem 4453* Lemma for onsucelsucexmid 4454. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5774), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4444. (Contributed by Jim Kingdon, 2-Aug-2019.)
{𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On

Theoremonsucelsucexmid 4454* The converse of onsucelsucr 4433 implies excluded middle. On the other hand, if 𝑦 is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4433 does hold, as seen at nnsucelsuc 6396. (Contributed by Jim Kingdon, 2-Aug-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)       (𝜑 ∨ ¬ 𝜑)

Theoremordsucunielexmid 4455* The converse of sucunielr 4435 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)

2.5  IZF Set Theory - add the Axiom of Set Induction

2.5.1  The ZF Axiom of Foundation would imply Excluded Middle

Theoremregexmidlemm 4456* Lemma for regexmid 4459. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       𝑦 𝑦𝐴

Theoremregexmidlem1 4457* Lemma for regexmid 4459. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       (∃𝑦(𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))

Theoremreg2exmidlema 4458* Lemma for reg2exmid 4460. If 𝐴 has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))

Theoremregexmid 4459* The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4461. (Contributed by Jim Kingdon, 3-Sep-2019.)

(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))       (𝜑 ∨ ¬ 𝜑)

Theoremreg2exmid 4460* If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝑧(∃𝑤 𝑤𝑧 → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)

2.5.2  Introduce the Axiom of Set Induction

Axiomax-setind 4461* Axiom of -Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory.

For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.)

(∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)

Theoremsetindel 4462* -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
(∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)

Theoremsetind 4463* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)

Theoremsetind2 4464 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
(𝒫 𝐴𝐴𝐴 = V)

Theoremelirr 4465 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.

The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4461, we could redefine Ord 𝐴 (df-iord 4297) to also require E Fr 𝐴 (df-frind 4263) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4466 (which under that definition would presumably not need ax-setind 4461 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4466. To encourage ordirr 4466 when possible, we mark this theorem as discouraged.

(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)

¬ 𝐴𝐴

Theoremordirr 4466 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4461. If in the definition of ordinals df-iord 4297, we also required that membership be well-founded on any ordinal (see df-frind 4263), then we could prove ordirr 4466 without ax-setind 4461. (Contributed by NM, 2-Jan-1994.)
(Ord 𝐴 → ¬ 𝐴𝐴)

Theoremonirri 4467 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On        ¬ 𝐴𝐴

Theoremnordeq 4468 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)

Theoremordn2lp 4469 An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))

Theoremorddisj 4470 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)

Theoremorddif 4471 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Theoremelirrv 4472 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
¬ 𝑥𝑥

Theoremsucprcreg 4473 A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Theoremruv 4474 The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
{𝑥𝑥𝑥} = V

TheoremruALT 4475 Alternate proof of Russell's Paradox ru 2913, simplified using (indirectly) the Axiom of Set Induction ax-setind 4461. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V

Theoremonprc 4476 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4411), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
¬ On ∈ V

Theoremsucon 4477 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On

Theoremen2lp 4478 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
¬ (𝐴𝐵𝐵𝐴)

Theorempreleq 4479 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremopthreg 4480 Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4461 (via the preleq 4479 step). See df-op 3542 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 = 𝐶𝐵 = 𝐷))

Theoremsuc11g 4481 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.)
((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Theoremsuc11 4482 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 𝐵𝐴 = 𝐵))

Theoremdtruex 4483* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4124 can also be summarized as "at least two sets exist", the difference is that dtruarb 4124 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

Theoremdtru 4484* 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 4483. (Contributed by Jim Kingdon, 29-Dec-2018.)
¬ ∀𝑥 𝑥 = 𝑦

Theoremeunex 4485 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Theoremordsoexmid 4486 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)

Theoremordsuc 4487 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)

Theoremonsucuni2 4488 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)

Theorem0elsucexmid 4489* 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)

Theoremnlimsucg 4490 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → ¬ Lim suc 𝐴)

Theoremordpwsucss 4491 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 4302 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4363) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4437).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4494). (Contributed by Jim Kingdon, 21-Jul-2019.)

(Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Theoremonnmin 4492 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 ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Theoremssnel 4493 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.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theoremordpwsucexmid 4494* The subset in ordpwsucss 4491 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)

Theoremordtri2or2exmid 4495* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

Theoremonintexmid 4496* 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 𝑥𝑦) → 𝑦𝑦)       (𝜑 ∨ ¬ 𝜑)

Theoremzfregfr 4497 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴

Theoremordfr 4498 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)

Theoremordwe 4499 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)

Theoremwetriext 4500* A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))       (𝜑𝐵 = 𝐶)

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