P. 46 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rabxfrd 4501*	Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
Theorem	rabxfr 4502*	Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
Theorem	reuhypd 4503*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	reuhyp 4504*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)    &   ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	uniexb 4505	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)
Theorem	pwexb 4506	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)
Theorem	elpwpwel 4507	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)
Theorem	univ 4508	The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
⊢ ∪ V = V
Theorem	eldifpw 4509	Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))
Theorem	op1stb 4510	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
Theorem	op1stbg 4511	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴)
Theorem	iunpw 4512*	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    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
Theorem	ifelpwung 4513	Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
Theorem	ifelpwund 4514	Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
Theorem	ifelpwun 4515	Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)
Theorem	ifexd 4516	Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
Theorem	ifexg 4517	Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Theorem	ifex 4518	Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) ∈ V
2.4.2  Ordinals (continued)
Theorem	ordon 4519	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
Theorem	ssorduni 4520	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)
Theorem	ssonuni 4521	The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
Theorem	ssonunii 4522	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)
Theorem	onun2 4523	The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
Theorem	onun2i 4524	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    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ On
Theorem	ordsson 4525	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 𝐴 → 𝐴 ⊆ On)
Theorem	onss 4526	An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
Theorem	onuni 4527	The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
Theorem	orduni 4528	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
⊢ (Ord 𝐴 → Ord ∪ 𝐴)
Theorem	bm2.5ii 4529*	Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	sucexb 4530	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Theorem	sucexg 4531	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	sucex 4532	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
Theorem	ordsucim 4533	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
⊢ (Ord 𝐴 → Ord suc 𝐴)
Theorem	onsuc 4534	The successor of an ordinal number is an ordinal number. Closed form of onsuci 4549. Forward implication of onsucb 4536. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
Theorem	ordsucg 4535	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
Theorem	onsucb 4536	A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4534. (Contributed by NM, 9-Sep-2003.)
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
Theorem	ordsucss 4537	The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
Theorem	ordelsuc 4538	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	onsucssi 4539	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	onsucmin 4540*	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	onsucelsucr 4541	Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4563. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6546. (Contributed by Jim Kingdon, 17-Jul-2019.)
⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))
Theorem	onsucsssucr 4542	The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4560. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
Theorem	sucunielr 4543	Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4564. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)
Theorem	unon 4544	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	onuniss2 4545*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ (𝐴 ∈ On → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)
Theorem	limon 4546	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
⊢ Lim On
Theorem	ordunisuc2r 4547*	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 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴))
Theorem	onssi 4548	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ 𝐴 ⊆ On
Theorem	onsuci 4549	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4534 and onsucb 4536. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ On
Theorem	onintonm 4550*	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 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)
Theorem	onintrab2im 4551	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)
Theorem	ordtriexmidlem 4552	Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4554 or weak linearity in ordsoexmid 4595) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
Theorem	ordtriexmidlem2 4553*	Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4554 or weak linearity in ordsoexmid 4595) with a proposition 𝜑. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
Theorem	ordtriexmid 4554*	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". Also see exmidontri 7301 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ontriexmidim 4555*	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4554. (Contributed by Jim Kingdon, 26-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
Theorem	ordtri2orexmid 4556*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	2ordpr 4557	Version of 2on 6480 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ Ord {∅, {∅}}
Theorem	ontr2exmid 4558*	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordtri2or2exmidlem 4559*	A set which is 2o if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Theorem	onsucsssucexmid 4560*	The converse of onsucsssucr 4542 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	onsucelsucexmidlem1 4561*	Lemma for onsucelsucexmid 4563. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
Theorem	onsucelsucexmidlem 4562*	Lemma for onsucelsucexmid 4563. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5910), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4552. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
Theorem	onsucelsucexmid 4563*	The converse of onsucelsucr 4541 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 4541 does hold, as seen at nnsucelsuc 6546. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordsucunielexmid 4564*	The converse of sucunielr 4543 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)    ⇒   ⊢ (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
Theorem	regexmidlemm 4565*	Lemma for regexmid 4568. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ ∃𝑦 𝑦 ∈ 𝐴
Theorem	regexmidlem1 4566*	Lemma for regexmid 4568. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))
Theorem	reg2exmidlema 4567*	Lemma for reg2exmid 4569. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))
Theorem	regexmid 4568*	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 4570. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	reg2exmid 4569*	If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.5.2  Introduce the Axiom of Set Induction
Axiom	ax-setind 4570*	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(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	setindel 4571*	∈-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V)
Theorem	setind 4572*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setind2 4573	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)
Theorem	elirr 4574	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 4570, we could redefine Ord 𝐴 (df-iord 4398) to also require E Fr 𝐴 (df-frind 4364) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4575 (which under that definition would presumably not need ax-setind 4570 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 4575. To encourage ordirr 4575 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.)
⊢ ¬ 𝐴 ∈ 𝐴
Theorem	ordirr 4575	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 4570. If in the definition of ordinals df-iord 4398, we also required that membership be well-founded on any ordinal (see df-frind 4364), then we could prove ordirr 4575 without ax-setind 4570. (Contributed by NM, 2-Jan-1994.)
⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
Theorem	onirri 4576	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ¬ 𝐴 ∈ 𝐴
Theorem	nordeq 4577	A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵)
Theorem	ordn2lp 4578	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 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
Theorem	orddisj 4579	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
Theorem	orddif 4580	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
Theorem	elirrv 4581	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
⊢ ¬ 𝑥 ∈ 𝑥
Theorem	sucprcreg 4582	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)
Theorem	ruv 4583	The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
Theorem	ruALT 4584	Alternate proof of Russell's Paradox ru 2985, simplified using (indirectly) the Axiom of Set Induction ax-setind 4570. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
Theorem	onprc 4585	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 4519), 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
Theorem	sucon 4586	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
⊢ suc On = On
Theorem	en2lp 4587	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.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
Theorem	preleq 4588	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    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthreg 4589	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4570 (via the preleq 4588 step). See df-op 3628 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 4590	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 4591	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 4592*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4221 can also be summarized as "at least two sets exist", the difference is that dtruarb 4221 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 4593*	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 4592. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	eunex 4594	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 4595	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 4596	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 4597	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 4598*	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 4599	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 4600	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 4403 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4464) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4545). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4603). (Contributed by Jim Kingdon, 21-Jul-2019.)
⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))