P. 359 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axextbdist 35801	axextb 2711 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
Theorem	19.12b 35802*	Version of 19.12vv 2349 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
Theorem	exnel 35803	There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦
Theorem	distel 35804	Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5442 and elirrv 9636.) (Contributed by Scott Fenton, 15-Dec-2010.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
Theorem	axextndbi 35805	axextnd 10631 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
21.11.14  Hypothesis builders
Theorem	hbntg 35806	A more general form of hbnt 2294. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Theorem	hbimtg 35807	A more general and closed form of hbim 2299. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))
Theorem	hbaltg 35808	A more general and closed form of hbal 2167. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	hbng 35809	A more general form of hbn 2295. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)
Theorem	hbimg 35810	A more general form of hbim 2299. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))
21.11.15  Well-founded zero, successor, and limits
Syntax	cwsuc 35811	Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)
Syntax	cwlim 35812	Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)
Definition	df-wsuc 35813	Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
Definition	df-wlim 35814*	Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
Theorem	wsuceq123 35815	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))
Theorem	wsuceq1 35816	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))
Theorem	wsuceq2 35817	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))
Theorem	wsuceq3 35818	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))
Theorem	nfwsuc 35819	Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    ⇒   ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)
Theorem	wlimeq12 35820	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Theorem	wlimeq1 35821	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))
Theorem	wlimeq2 35822	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))
Theorem	nfwlim 35823	Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)
Theorem	elwlim 35824	Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
Theorem	wzel 35825	The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)
Theorem	wsuclem 35826*	Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))
Theorem	wsucex 35827	Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Theorem	wsuccl 35828*	If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
Theorem	wsuclb 35829	A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    ⇒   ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Theorem	wlimss 35830	The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
⊢ WLim(𝑅, 𝐴) ⊆ 𝐴
21.11.16  Quantifier-free definitions
Syntax	ctxp 35831	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 35832	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 35833	Declare the subset relationship class.
class SSet
Syntax	ctrans 35834	Declare the transitive set class.
class Trans
Syntax	cbigcup 35835	Declare the set union relationship.
class Bigcup
Syntax	cfix 35836	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 35837	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 35838	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 35839	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 35840	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 35841	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 35842	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 35843	Declare the syntax for the image function.
class Img
Syntax	cdomain 35844	Declare the syntax for the domain function.
class Domain
Syntax	crange 35845	Declare the syntax for the range function.
class Range
Syntax	capply 35846	Declare the syntax for the application function.
class Apply
Syntax	ccup 35847	Declare the syntax for the cup function.
class Cup
Syntax	ccap 35848	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 35849	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 35850	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 35851	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 35852	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 35853	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 35854	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 35855	Define the tail cross of two classes. Membership in this class is defined by txpss3v 35879 and brtxp 35881. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 35856	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 35885 and brpprod 35886. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 35857	Define the subset class. For the value, see brsset 35890. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 35858	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 35859	Define the Bigcup function, which, per fvbigcup 35903, 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 35860	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 35861	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 35862	Define the class of all functions. See elfuns 35916 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 35863	Define the singleton function. See brsingle 35918 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 35864	Define the class of all singletons. See elsingles 35919 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 35865	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 35931 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 35866	Define the cartesian product function. See brcart 35933 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 35867	Define the image function. See brimg 35938 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 35868	Define the domain function. See brdomain 35934 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 35869	Define the range function. See brrange 35935 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 35870	Define the little cup function. See brcup 35940 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 35871	Define the little cap function. See brcap 35941 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 35872	Define the restriction function. See brrestrict 35950 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
Definition	df-succf 35873	Define the successor function. See brsuccf 35942 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
Definition	df-apply 35874	Define the application function. See brapply 35939 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 35875	Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 35944 and funpartfv 35946 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
Definition	df-fullfun 35876	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 35877	Define the upper bound relationship functor. See brub 35955 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
Definition	df-lb 35878	Define the lower bound relationship functor. See brlb 35956 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ LB𝑅 = UB◡𝑅
Theorem	txpss3v 35879	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 35880	A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⊗ 𝐵)
Theorem	brtxp 35881	Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 35879, 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 35882*	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 35883	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 35884	A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
Theorem	pprodss4v 35885	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 35886	Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 35885, 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 35887*	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 35888*	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 35889	The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel SSet
Theorem	brsset 35890	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 35891	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 35892	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 35893	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 35894	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
Theorem	brtxpsd 35895*	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 35896*	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 35897*	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 35898	The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel Bigcup
Theorem	brbigcup 35899	Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)
Theorem	dfbigcup2 35900	Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)