P. 343 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	idsset 34201	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 34202	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 34203	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 34204	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
Theorem	brtxpsd 34205*	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 34206*	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 34207*	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 34208	The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel Bigcup
Theorem	brbigcup 34209	Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)
Theorem	dfbigcup2 34210	Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
Theorem	fobigcup 34211	Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup :V–onto→V
Theorem	fnbigcup 34212	Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Bigcup Fn V
Theorem	fvbigcup 34213	For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴
Theorem	elfix 34214	Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	elfix2 34215	Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	dffix2 34216	The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 = ran (𝐴 ∩ I )
Theorem	fixssdm 34217	The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ dom 𝐴
Theorem	fixssrn 34218	The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ ran 𝐴
Theorem	fixcnv 34219	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 34220	The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)
Theorem	ellimits 34221	Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴)
Theorem	limitssson 34222	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 34223	A quantifier-free definition of ω that does not depend on ax-inf 9405. (Note: label was changed from dfom5 9417 to dfom5b 34223 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ ω = (On ∩ ∩ Limits )
Theorem	sscoid 34224	A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))
Theorem	dffun10 34225	Another potential definition of functionality. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
⊢ (Fun 𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
Theorem	elfuns 34226	Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ Funs ↔ Fun 𝐹)
Theorem	elfunsg 34227	Closed form of elfuns 34226. (Contributed by Scott Fenton, 2-May-2014.)
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹))
Theorem	brsingle 34228	The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})
Theorem	elsingles 34229*	Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	fnsingle 34230	The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Singleton Fn V
Theorem	fvsingle 34231	The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
⊢ (Singleton‘𝐴) = {𝐴}
Theorem	dfsingles2 34232*	Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
Theorem	snelsingles 34233	A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ Singletons
Theorem	dfiota3 34234	A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons )
Theorem	dffv5 34235	Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )
Theorem	unisnif 34236	Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
Theorem	brimage 34237	Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))
Theorem	brimageg 34238	Closed form of brimage 34237. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))
Theorem	funimage 34239	Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun Image𝐴
Theorem	fnimage 34240*	Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
Theorem	imageval 34241*	The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
Theorem	fvimage 34242	Value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))
Theorem	brcart 34243	Binary relation form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))
Theorem	brdomain 34244	Binary relation form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)
Theorem	brrange 34245	Binary relation form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)
Theorem	brdomaing 34246	Closed form of brdomain 34244. (Contributed by Scott Fenton, 2-May-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))
Theorem	brrangeg 34247	Closed form of brrange 34245. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))
Theorem	brimg 34248	Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Img𝐶 ↔ 𝐶 = (𝐴 “ 𝐵))
Theorem	brapply 34249	Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵))
Theorem	brcup 34250	Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵))
Theorem	brcap 34251	Binary relation form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵))
Theorem	brsuccf 34252	Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)
Theorem	funpartlem 34253*	Lemma for funpartfun 34254. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Theorem	funpartfun 34254	The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ Fun Funpart𝐹
Theorem	funpartss 34255	The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Funpart𝐹 ⊆ 𝐹
Theorem	funpartfv 34256	The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)
Theorem	fullfunfnv 34257	The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ FullFun𝐹 Fn V
Theorem	fullfunfv 34258	The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
Theorem	brfullfun 34259	A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))
Theorem	brrestrict 34260	Binary relation form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Restrict𝐶 ↔ 𝐶 = (𝐴 ↾ 𝐵))
Theorem	dfrecs2 34261	A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)
⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))
Theorem	dfrdg4 34262	A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
Theorem	dfint3 34263	Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴))
Theorem	imagesset 34264	The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
⊢ Image◡ SSet ⊆ SSet
Theorem	brub 34265*	Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
⊢ 𝑆 ∈ V    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
Theorem	brlb 34266*	Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
⊢ 𝑆 ∈ V    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
20.9.36  Alternate ordered pairs
Syntax	caltop 34267	Declare the syntax for an alternate ordered pair.
class ⟪𝐴, 𝐵⟫
Syntax	caltxp 34268	Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)
Definition	df-altop 34269	An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 34280), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
Definition	df-altxp 34270*	Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
Theorem	altopex 34271	Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ⟪𝐴, 𝐵⟫ ∈ V
Theorem	altopthsn 34272	Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
Theorem	altopeq12 34273	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
Theorem	altopeq1 34274	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
Theorem	altopeq2 34275	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)
Theorem	altopth1 34276	Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))
Theorem	altopth2 34277	Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐵 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))
Theorem	altopthg 34278	Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	altopthbg 34279	Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	altopth 34280	The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 5392), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altopthb 34281	Alternate ordered pair theorem with different sethood requirements. See altopth 34280 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altopthc 34282	Alternate ordered pair theorem with different sethood requirements. See altopth 34280 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altopthd 34283	Alternate ordered pair theorem with different sethood requirements. See altopth 34280 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altxpeq1 34284	Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶))
Theorem	altxpeq2 34285	Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵))
Theorem	elaltxp 34286*	Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Theorem	altopelaltxp 34287	Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5626, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
Theorem	altxpsspw 34288	An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Theorem	altxpexg 34289	The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V)
Theorem	rankaltopb 34290	Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
Theorem	nfaltop 34291	Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟪𝐴, 𝐵⟫
Theorem	sbcaltop 34292*	Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
20.9.37  Geometry in the Euclidean space
20.9.37.1  Congruence properties
Syntax	cofs 34293	Declare the syntax for the outer five segment configuration.
class OuterFiveSeg
Definition	df-ofs 34294*	The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 27315). See brofs 34316 and 5segofs 34317 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
⊢ OuterFiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑥, 𝑦⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑦, 𝑧⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑦, 𝑤⟩)))}
Theorem	cgrrflx2d 34295	Deduction form of axcgrrflx 27291. (Contributed by Scott Fenton, 13-Oct-2013.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)
Theorem	cgrtr4d 34296	Deduction form of axcgrtr 27292. (Contributed by Scott Fenton, 13-Oct-2013.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   ⊢ (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)
Theorem	cgrtr4and 34297	Deduction form of axcgrtr 27292. (Contributed by Scott Fenton, 13-Oct-2013.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)
Theorem	cgrrflx 34298	Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)
Theorem	cgrrflxd 34299	Deduction form of cgrrflx 34298. (Contributed by Scott Fenton, 13-Oct-2013.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)
Theorem	cgrcomim 34300	Congruence commutes on the two sides. Implication version. Theorem 2.2 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝐵⟩))