P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	etasslt 32801*	A restatement of noeta 32749 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
Theorem	scutbdaybnd 32802	An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
Theorem	scutbdaylt 32803	If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)
⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋))
Theorem	slerec 32804*	A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))
Theorem	sltrec 32805*	A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
20.9.29  Surreal numbers - cuts and options
Syntax	cmade 32806	Declare the symbol for the made by function.
class M
Syntax	cold 32807	Declare the symbol for the older than function.
class O
Syntax	cnew 32808	Declare the symbol for the new on function.
class N
Syntax	cleft 32809	Declare the symbol for the left option function.
class L
Syntax	cright 32810	Declare the symbol for the right option function.
class R
Definition	df-made 32811	Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
Definition	df-old 32812	Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
Definition	df-new 32813	Define the newer than function. This function carries an ordinal to all surreals made on that day. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ N = (𝑥 ∈ On ↦ (( O ‘𝑥) ∖ ( M ‘𝑥)))
Definition	df-left 32814*	Define the left options of a surreal. This is the set of surreals that are "closest" on the left to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑦 <s 𝑧 ∧ 𝑧 <s 𝑥) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))})
Definition	df-right 32815*	Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))})
Theorem	madeval 32816	The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
Theorem	madeval2 32817*	Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
20.9.30  Quantifier-free definitions
Syntax	ctxp 32818	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 32819	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 32820	Declare the subset relationship class.
class SSet
Syntax	ctrans 32821	Declare the transitive set class.
class Trans
Syntax	cbigcup 32822	Declare the set union relationship.
class Bigcup
Syntax	cfix 32823	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 32824	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 32825	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 32826	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 32827	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 32828	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 32829	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 32830	Declare the syntax for the image function.
class Img
Syntax	cdomain 32831	Declare the syntax for the domain function.
class Domain
Syntax	crange 32832	Declare the syntax for the range function.
class Range
Syntax	capply 32833	Declare the syntax for the application function.
class Apply
Syntax	ccup 32834	Declare the syntax for the cup function.
class Cup
Syntax	ccap 32835	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 32836	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 32837	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 32838	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 32839	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 32840	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 32841	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 32842	Define the tail cross of two classes. Membership in this class is defined by txpss3v 32866 and brtxp 32868. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 32843	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 32872 and brpprod 32873. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 32844	Define the subset class. For the value, see brsset 32877. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 32845	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 32846	Define the Bigcup function, which, per fvbigcup 32890, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
Definition	df-fix 32847	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 32848	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 32849	Define the class of all functions. See elfuns 32903 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 32850	Define the singleton function. See brsingle 32905 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 32851	Define the class of all singletons. See elsingles 32906 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 32852	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 32918 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 32853	Define the cartesian product function. See brcart 32920 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 32854	Define the image function. See brimg 32925 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 32855	Define the domain function. See brdomain 32921 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 32856	Define the range function. See brrange 32922 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 32857	Define the little cup function. See brcup 32927 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V)))
Definition	df-cap 32858	Define the little cap function. See brcap 32928 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗ V)))
Definition	df-restrict 32859	Define the restriction function. See brrestrict 32937 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
Definition	df-succf 32860	Define the successor function. See brsuccf 32929 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
Definition	df-apply 32861	Define the application function. See brapply 32926 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
Definition	df-funpart 32862	Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 32931 and funpartfv 32933 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
Definition	df-fullfun 32863	Define the full function over 𝐹. This is a function with domain V that always agrees with 𝐹 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
Definition	df-ub 32864	Define the upper bound relationship functor. See brub 32942 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
Definition	df-lb 32865	Define the lower bound relationship functor. See brlb 32943 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ LB𝑅 = UB◡𝑅
Theorem	txpss3v 32866	A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))
Theorem	txprel 32867	A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⊗ 𝐵)
Theorem	brtxp 32868	Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32866, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
Theorem	brtxp2 32869*	The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
Theorem	dfpprod2 32870	Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
Theorem	pprodcnveq 32871	A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
Theorem	pprodss4v 32872	The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
Theorem	brpprod 32873	Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32872, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    &   ⊢ 𝑊 ∈ V    ⇒   ⊢ (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊))
Theorem	brpprod3a 32874*	Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
Theorem	brpprod3b 32875*	Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧∃𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍))
Theorem	relsset 32876	The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel SSet
Theorem	brsset 32877	For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵)
Theorem	idsset 32878	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 32879	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 32880	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 32881	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
Theorem	brtxpsd 32882*	Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴))
Theorem	brtxpsd2 32883*	Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   ⊢ 𝐴𝐶𝐵    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴))
Theorem	brtxpsd3 32884*	A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   ⊢ 𝐴𝐶𝐵    &   ⊢ (𝑥 ∈ 𝑋 ↔ 𝑥𝑆𝐴)    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋)
Theorem	relbigcup 32885	The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel Bigcup
Theorem	brbigcup 32886	Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)
Theorem	dfbigcup2 32887	Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
Theorem	fobigcup 32888	Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup :V–onto→V
Theorem	fnbigcup 32889	Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Bigcup Fn V
Theorem	fvbigcup 32890	For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴
Theorem	elfix 32891	Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	elfix2 32892	Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	dffix2 32893	The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 = ran (𝐴 ∩ I )
Theorem	fixssdm 32894	The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ dom 𝐴
Theorem	fixssrn 32895	The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ ran 𝐴
Theorem	fixcnv 32896	The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 = Fix ◡𝐴
Theorem	fixun 32897	The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)
Theorem	ellimits 32898	Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴)
Theorem	limitssson 32899	The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits ⊆ On
Theorem	dfom5b 32900	A quantifier-free definition of ω that does not depend on ax-inf 8895. (Note: label was changed from dfom5 8907 to dfom5b 32900 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ ω = (On ∩ ∩ Limits )