P. 320 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ssltex1 31901	The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
Theorem	ssltex2 31902	The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
Theorem	ssltss1 31903	The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
Theorem	ssltss2 31904	The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
Theorem	ssltsep 31905*	The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
Theorem	sssslt1 31906	Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)
Theorem	sssslt2 31907	Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)
Theorem	nulsslt 31908	The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Theorem	nulssgt 31909	The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)
Theorem	conway 31910*	Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
Theorem	scutval 31911*	The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
Theorem	scutcut 31912	Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
Theorem	scutbday 31913*	The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
Theorem	sslttr 31914	Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 <<s 𝐶)
Theorem	ssltun1 31915	Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)
Theorem	ssltun2 31916	Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))
Theorem	scutun12 31917	Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))
Theorem	dmscut 31918	The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ dom |s = <<s
Theorem	scutf 31919	Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
⊢ |s : <<s ⟶ No
Theorem	etasslt 31920*	A restatement of noeta 31868 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 31921	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 31922	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 31923*	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 31924*	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)))
20.8.29  Surreal numbers - cuts and options
Syntax	cmade 31925	Declare the symbol for the made by function.
class M
Syntax	cold 31926	Declare the symbol for the older than function.
class O
Syntax	cnew 31927	Declare the symbol for the new on function.
class N
Syntax	cleft 31928	Declare the symbol for the left option function.
class L
Syntax	cright 31929	Declare the symbol for the right option function.
class R
Definition	df-made 31930	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 31931	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 31932	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 31933*	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 31934*	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 31935	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 31936*	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)})
20.8.30  Quantifier-free definitions
Syntax	ctxp 31937	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 31938	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 31939	Declare the subset relationship class.
class SSet
Syntax	ctrans 31940	Declare the transitive set class.
class Trans
Syntax	cbigcup 31941	Declare the set union relationship.
class Bigcup
Syntax	cfix 31942	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 31943	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 31944	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 31945	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 31946	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 31947	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 31948	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 31949	Declare the syntax for the image function.
class Img
Syntax	cdomain 31950	Declare the syntax for the domain function.
class Domain
Syntax	crange 31951	Declare the syntax for the range function.
class Range
Syntax	capply 31952	Declare the syntax for the application function.
class Apply
Syntax	ccup 31953	Declare the syntax for the cup function.
class Cup
Syntax	ccap 31954	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 31955	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 31956	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 31957	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 31958	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 31959	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 31960	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 31961	Define the tail cross of two classes. Membership in this class is defined by txpss3v 31985 and brtxp 31987. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 31962	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 31991 and brpprod 31992. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 31963	Define the subset class. For the value, see brsset 31996. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 31964	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 31965	Define the Bigcup function, which, per fvbigcup 32009, 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 31966	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 31967	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 31968	Define the class of all functions. See elfuns 32022 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 31969	Define the singleton function. See brsingle 32024 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 31970	Define the class of all singletons. See elsingles 32025 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 31971	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 32037 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 31972	Define the cartesian product function. See brcart 32039 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 31973	Define the image function. See brimg 32044 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 31974	Define the domain function. See brdomain 32040 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 31975	Define the range function. See brrange 32041 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 31976	Define the little cup function. See brcup 32046 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 31977	Define the little cap function. See brcap 32047 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 31978	Define the restriction function. See brrestrict 32056 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
Definition	df-succf 31979	Define the successor function. See brsuccf 32048 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
Definition	df-apply 31980	Define the application function. See brapply 32045 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 31981	Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 32050 and funpartfv 32052 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
Definition	df-fullfun 31982	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 31983	Define the upper bound relationship functor. See brub 32061 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
Definition	df-lb 31984	Define the lower bound relationship functor. See brlb 32062 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ LB𝑅 = UB◡𝑅
Theorem	txpss3v 31985	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 31986	A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⊗ 𝐵)
Theorem	brtxp 31987	Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 31985, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
Theorem	brtxp2 31988*	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 31989	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 31990	A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
Theorem	pprodss4v 31991	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.)
⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
Theorem	brpprod 31992	Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 31991, 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 31993*	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 31994*	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 31995	The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel SSet
Theorem	brsset 31996	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 31997	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 31998	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 31999	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 32000	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))