P. 364 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axextdist 36301	ax-ext 2738 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Theorem	axextbdist 36302	axextb 2741 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 36303*	Version of 19.12vv 2382 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 36304	There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦
Theorem	distel 36305	Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5422 and elirrv 9561.) (Contributed by Scott Fenton, 15-Dec-2010.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
Theorem	axextndbi 36306	axextnd 10586 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
21.11.13  Hypothesis builders
Theorem	hbntg 36307	A more general form of hbnt 2332. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Theorem	hbimtg 36308	A more general and closed form of hbim 2337. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))
Theorem	hbaltg 36309	A more general and closed form of hbal 2205. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	hbng 36310	A more general form of hbn 2333. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)
Theorem	hbimg 36311	A more general form of hbim 2337. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))
21.11.14  Well-founded zero, successor, and limits
Syntax	cwsuc 36312	Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)
Syntax	cwlim 36313	Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)
Definition	df-wsuc 36314	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 36315*	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 36316	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 36317	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 36318	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 36319	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 36320	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 36321	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 36322	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))
Theorem	wlimeq2 36323	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))
Theorem	nfwlim 36324	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 36325	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 36326	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 36327*	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 36328	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 36329*	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 36330	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 36331	The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
⊢ WLim(𝑅, 𝐴) ⊆ 𝐴
21.11.15  Quantifier-free definitions
Syntax	ctxp 36332	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 36333	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 36334	Declare the subset relationship class.
class SSet
Syntax	ctrans 36335	Declare the transitive set class.
class Trans
Syntax	cbigcup 36336	Declare the set union relationship.
class Bigcup
Syntax	cfix 36337	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 36338	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 36339	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 36340	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 36341	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 36342	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 36343	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 36344	Declare the syntax for the image function.
class Img
Syntax	cdomain 36345	Declare the syntax for the domain function.
class Domain
Syntax	crange 36346	Declare the syntax for the range function.
class Range
Syntax	capply 36347	Declare the syntax for the application function.
class Apply
Syntax	ccup 36348	Declare the syntax for the cup function.
class Cup
Syntax	ccap 36349	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 36350	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 36351	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 36352	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 36353	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 36354	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 36355	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 36356	Define the tail cross of two classes. Membership in this class is defined by txpss3v 36380 and brtxp 36382. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 36357	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 36386 and brpprod 36387. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 36358	Define the subset class. For the value, see brsset 36391. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 36359	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 36360	Define the Bigcup function, which, per fvbigcup 36404, 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 36361	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 36362	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 36363	Define the class of all functions. See elfuns 36417 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 36364	Define the singleton function. See brsingle 36419 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 36365	Define the class of all singletons. See elsingles 36420 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 36366	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 36432 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 36367	Define the cartesian product function. See brcart 36434 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 36368	Define the image function. See brimg 36439 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 36369	Define the domain function. See brdomain 36435 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 36370	Define the range function. See brrange 36436 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 36371	Define the little cup function. See brcup 36441 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 36372	Define the little cap function. See brcap 36442 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 36373	Define the restriction function. See brrestrict 36453 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
Definition	df-succf 36374	Define the successor function. See its alternate version dfsuccf2 36445. See brsuccf 36444 for its value. Cf. the equivalent df-sucmap 39143 family. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
Definition	df-apply 36375	Define the application function. See brapply 36440 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 36376	Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 36447 and funpartfv 36449 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
Definition	df-fullfun 36377	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 36378	Define the upper bound relationship functor. See brub 36458 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
Definition	df-lb 36379	Define the lower bound relationship functor. See brlb 36459 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ LB𝑅 = UB◡𝑅
Theorem	txpss3v 36380	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 36381	A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⊗ 𝐵)
Theorem	brtxp 36382	Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 36380, 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 36383*	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 36384	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 36385	A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
Theorem	pprodss4v 36386	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 36387	Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 36386, 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 36388*	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 36389*	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 36390	The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel SSet
Theorem	brsset 36391	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 36392	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 36393	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 36394	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 36395	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
Theorem	brtxpsd 36396*	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 36397*	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 36398*	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 36399	The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel Bigcup
Theorem	brbigcup 36400	Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)