P. 44 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	freq2 4301	Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
|- ( A = B -> ( R Fr A <-> R Fr B ) )
Theorem	frforeq3 4302	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( S = T -> (FrFor R A S <-> FrFor R A T ) )
Theorem	nffrfor 4303	Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x R   &    |- F/_ x A   &    |- F/_ x S   =>    |- F/ xFrFor R A S
Theorem	nffr 4304	Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x R   &    |- F/_ x A   =>    |- F/ x R Fr A
Theorem	frirrg 4305	A well-founded relation is irreflexive. This is the case where A exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
|- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B )
Theorem	fr0 4306	Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
|- R Fr (/)
Theorem	frind 4307*	Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ( ch /\ x e. A ) -> ( A. y e. A ( y R x -> ps ) -> ph ) )   &    |- ( ch -> R Fr A )   &    |- ( ch -> A e. V )   =>    |- ( ( ch /\ x e. A ) -> ph )
Theorem	efrirr 4308	Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
|- ( _E Fr A -> -. A e. A )
Theorem	tz7.2 4309	Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent _E Fr A. (Contributed by NM, 4-May-1994.)
|- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )
Theorem	nfwe 4310	Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x R   &    |- F/_ x A   =>    |- F/ x R We A
Theorem	weeq1 4311	Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
|- ( R = S -> ( R We A <-> S We A ) )
Theorem	weeq2 4312	Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
|- ( A = B -> ( R We A <-> R We B ) )
Theorem	wefr 4313	A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
|- ( R We A -> R Fr A )
Theorem	wepo 4314	A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
|- ( ( R We A /\ A e. V ) -> R Po A )
Theorem	wetrep 4315*	An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
|- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x e. y /\ y e. z ) -> x e. z ) )
Theorem	we0 4316	Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
|- R We (/)
2.3.10  Ordinals
Syntax	word 4317	Extend the definition of a wff to include the ordinal predicate.
wff Ord A
Syntax	con0 4318	Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)
class On
Syntax	wlim 4319	Extend the definition of a wff to include the limit ordinal predicate.
wff Lim A
Syntax	csuc 4320	Extend class notation to include the successor function.
class suc A
Definition	df-iord 4321*	Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4322 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
Theorem	dford3 4322*	Alias for df-iord 4321. Use it instead of df-iord 4321 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
|- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
Definition	df-on 4323	Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
|- On = { x | Ord x }
Definition	df-ilim 4324	Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes A =/= (/) to (/) e. A (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4325 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Theorem	dflim2 4325	Alias for df-ilim 4324. Use it instead of df-ilim 4324 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
Definition	df-suc 4326	Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4367). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)
|- suc A = ( A u. { A } )
Theorem	ordeq 4327	Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Ord A <-> Ord B ) )
Theorem	elong 4328	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> ( A e. On <-> Ord A ) )
Theorem	elon 4329	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
|- A e. _V   =>    |- ( A e. On <-> Ord A )
Theorem	eloni 4330	An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
|- ( A e. On -> Ord A )
Theorem	elon2 4331	An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
|- ( A e. On <-> ( Ord A /\ A e. _V ) )
Theorem	limeq 4332	Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A = B -> ( Lim A <-> Lim B ) )
Theorem	ordtr 4333	An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> Tr A )
Theorem	ordelss 4334	An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
|- ( ( Ord A /\ B e. A ) -> B C_ A )
Theorem	trssord 4335	A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
|- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )
Theorem	ordelord 4336	An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
|- ( ( Ord A /\ B e. A ) -> Ord B )
Theorem	tron 4337	The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
|- Tr On
Theorem	ordelon 4338	An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
|- ( ( Ord A /\ B e. A ) -> B e. On )
Theorem	onelon 4339	An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
|- ( ( A e. On /\ B e. A ) -> B e. On )
Theorem	ordin 4340	The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
|- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
Theorem	onin 4341	The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
|- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On )
Theorem	onelss 4342	An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A e. On -> ( B e. A -> B C_ A ) )
Theorem	ordtr1 4343	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
|- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )
Theorem	ontr1 4344	Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
|- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) )
Theorem	onintss 4345*	If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) )
Theorem	ord0 4346	The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
|- Ord (/)
Theorem	0elon 4347	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
|- (/) e. On
Theorem	inton 4348	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
|- |^| On = (/)
Theorem	nlim0 4349	The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- -. Lim (/)
Theorem	limord 4350	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
|- ( Lim A -> Ord A )
Theorem	limuni 4351	A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
|- ( Lim A -> A = U. A )
Theorem	limuni2 4352	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
|- ( Lim A -> Lim U. A )
Theorem	0ellim 4353	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
|- ( Lim A -> (/) e. A )
Theorem	limelon 4354	A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
|- ( ( A e. B /\ Lim A ) -> A e. On )
Theorem	onn0 4355	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
|- On =/= (/)
Theorem	onm 4356	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
|- E. x x e. On
Theorem	suceq 4357	Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A = B -> suc A = suc B )
Theorem	elsuci 4358	Membership in a successor. This one-way implication does not require that either A or B be sets. (Contributed by NM, 6-Jun-1994.)
|- ( A e. suc B -> ( A e. B \/ A = B ) )
Theorem	elsucg 4359	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
|- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc2g 4360	Variant of membership in a successor, requiring that B rather than A be a set. (Contributed by NM, 28-Oct-2003.)
|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc 4361	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( A e. suc B <-> ( A e. B \/ A = B ) )
Theorem	elsuc2 4362	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( B e. suc A <-> ( B e. A \/ B = A ) )
Theorem	nfsuc 4363	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
|- F/_ x A   =>    |- F/_ x suc A
Theorem	elelsuc 4364	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
|- ( A e. B -> A e. suc B )
Theorem	sucel 4365*	Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
|- ( suc A e. B <-> E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
Theorem	suc0 4366	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
|- suc (/) = { (/) }
Theorem	sucprc 4367	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
|- ( -. A e. _V -> suc A = A )
Theorem	unisuc 4368	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- ( Tr A <-> U. suc A = A )
Theorem	unisucg 4369	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.)
|- ( A e. V -> ( Tr A <-> U. suc A = A ) )
Theorem	sssucid 4370	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
|- A C_ suc A
Theorem	sucidg 4371	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- ( A e. V -> A e. suc A )
Theorem	sucid 4372	A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- A e. _V   =>    |- A e. suc A
Theorem	nsuceq0g 4373	No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
|- ( A e. V -> suc A =/= (/) )
Theorem	eqelsuc 4374	A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
|- A e. _V   =>    |- ( A = B -> A e. suc B )
Theorem	iunsuc 4375*	Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. suc A B = ( U_ x e. A B u. C )
Theorem	suctr 4376	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
|- ( Tr A -> Tr suc A )
Theorem	trsuc 4377	A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( ( Tr A /\ suc B e. A ) -> B e. A )
Theorem	trsucss 4378	A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
|- ( Tr A -> ( B e. suc A -> B C_ A ) )
Theorem	sucssel 4379	A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
|- ( A e. V -> ( suc A C_ B -> A e. B ) )
Theorem	orduniss 4380	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
|- ( Ord A -> U. A C_ A )
Theorem	onordi 4381	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Ord A
Theorem	ontrci 4382	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Tr A
Theorem	oneli 4383	A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B e. On )
Theorem	onelssi 4384	A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- ( B e. A -> B C_ A )
Theorem	onelini 4385	An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B = ( B i^i A ) )
Theorem	oneluni 4386	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> ( A u. B ) = A )
Theorem	onunisuci 4387	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- U. suc A = A
2.4  IZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union
Axiom	ax-un 4388*	Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set y exists that includes the union of a given set x i.e. the collection of all members of the members of x. The variant axun2 4390 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4391. A version using class notation is uniex 4392. This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4081), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264). The union of a class df-uni 3769 should not be confused with the union of two classes df-un 3102. Their relationship is shown in unipr 3782. (Contributed by NM, 23-Dec-1993.)
|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
Theorem	zfun 4389*	Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )
Theorem	axun2 4390*	A variant of the Axiom of Union ax-un 4388. For any set x, there exists a set y whose members are exactly the members of the members of x i.e. the union of x. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
Theorem	uniex2 4391*	The Axiom of Union using the standard abbreviation for union. Given any set x, its union y exists. (Contributed by NM, 4-Jun-2006.)
|- E. y y = U. x
Theorem	uniex 4392	The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2715), then the union of A is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
|- A e. _V   =>    |- U. A e. _V
Theorem	vuniex 4393	The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
|- U. x e. _V
Theorem	uniexg 4394	The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A e. V instead of A e. _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant _V is one possible substitution for class variable V. (Contributed by NM, 25-Nov-1994.)
|- ( A e. V -> U. A e. _V )
Theorem	unex 4395	The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
|- A e. _V   &    |- B e. _V   =>    |- ( A u. B ) e. _V
Theorem	unexb 4396	Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
|- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )
Theorem	unexg 4397	A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )
Theorem	tpexg 4398	An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> { A , B , C } e. _V )
Theorem	unisn3 4399*	Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
|- ( A e. B -> U. { x e. B | x = A } = A )
Theorem	abnexg 4400*	Sufficient condition for a class abstraction to be a proper class. The class F can be thought of as an expression in x and the abstraction appearing in the statement as the class of values F as x varies through A. Assuming the antecedents, if that class is a set, then so is the "domain" A. The converse holds without antecedent, see abrexexg 6056. Note that the second antecedent A. x e. A x e. F cannot be translated to A C_ F since F may depend on x. In applications, one may take F = { x } or F = ~P x (see snnex 4402 and pwnex 4403 respectively, proved from abnex 4401, which is a consequence of abnexg 4400 with A = _V). (Contributed by BJ, 2-Dec-2021.)
|- ( A. x e. A ( F e. V /\ x e. F ) -> ( { y | E. x e. A y = F } e. W -> A e. _V ) )