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Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexmid0el 4201 Excluded middle is equivalent to decidability of  (/) being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. xDECID  (/)  e.  x )
 
Theoremexmidel 4202* Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x A. yDECID  x  e.  y )
 
Theoremexmidundif 4203* Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3503 and undifdcss 6915 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x A. y ( x  C_  y  <->  ( x  u.  ( y  \  x ) )  =  y ) )
 
Theoremexmidundifim 4204* Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4203 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
 |-  (EXMID  <->  A. x A. y ( x  C_  y  ->  ( x  u.  ( y 
 \  x ) )  =  y ) )
 
2.3.3  Axiom of Pairing
 
Axiomax-pr 4205* The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4121). (Contributed by NM, 14-Nov-2006.)
 |- 
 E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
 
Theoremzfpair2 4206 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4205. (Contributed by NM, 14-Nov-2006.)
 |- 
 { x ,  y }  e.  _V
 
Theoremprexg 4207 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3701, prprc1 3699, and prprc2 3700. (Contributed by Jim Kingdon, 16-Sep-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
 
Theoremsnelpwi 4208 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
Theoremsnelpw 4209 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  e.  ~P B )
 
Theoremprelpwi 4210 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C )
 
Theoremrext 4211* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
 |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
 
Theoremsspwb 4212 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 |-  ( A  C_  B  <->  ~P A  C_  ~P B )
 
Theoremunipw 4213 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
 |- 
 U. ~P A  =  A
 
Theorempwel 4214 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )
 
Theorempwtr 4215 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
 |-  ( Tr  A  <->  Tr  ~P A )
 
Theoremssextss 4216* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
 |-  ( A  C_  B  <->  A. x ( x  C_  A  ->  x  C_  B ) )
 
Theoremssext 4217* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
 |-  ( A  =  B  <->  A. x ( x  C_  A 
 <->  x  C_  B )
 )
 
Theoremnssssr 4218* Negation of subclass relationship. Compare nssr 3215. (Contributed by Jim Kingdon, 17-Sep-2018.)
 |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A  C_  B )
 
Theorempweqb 4219 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 |-  ( A  =  B  <->  ~P A  =  ~P B )
 
Theoremintid 4220* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
 |-  A  e.  _V   =>    |-  |^| { x  |  A  e.  x }  =  { A }
 
Theoremeuabex 4221 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
 |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
 
Theoremmss 4222* An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
 |-  ( E. y  y  e.  A  ->  E. x ( x  C_  A  /\  E. z  z  e.  x ) )
 
Theoremexss 4223* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
 |-  ( E. x  e.  A  ph  ->  E. y
 ( y  C_  A  /\  E. x  e.  y  ph ) )
 
Theoremopexg 4224 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  <. A ,  B >.  e.  _V )
 
Theoremopex 4225 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  e. 
 _V
 
Theoremotexg 4226 An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
 |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  W ) 
 ->  <. A ,  B ,  C >.  e.  _V )
 
Theoremelop 4227 An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  e.  <. B ,  C >. 
 <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
 
Theoremopi1 4228 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { A }  e.  <. A ,  B >.
 
Theoremopi2 4229 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { A ,  B }  e.  <. A ,  B >.
 
2.3.4  Ordered pair theorem
 
Theoremopm 4230* An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( E. x  x  e.  <. A ,  B >.  <-> 
 ( A  e.  _V  /\  B  e.  _V )
 )
 
Theoremopnzi 4231 An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4230). (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  =/=  (/)
 
Theoremopth1 4232 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  A  =  C )
 
Theoremopth 4233 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that  C and  D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremopthg 4234 Ordered pair theorem.  C and  D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopthg2 4235 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopth2 4236 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremotth2 4237 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <.
 <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
 
Theoremotth 4238 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
 )
 
Theoremeqvinop 4239* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. ) )
 
Theoremcopsexg 4240* Substitution of class  A for ordered pair  <. x ,  y
>.. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 ) )
 
Theoremcopsex2t 4241* Closed theorem form of copsex2g 4242. (Contributed by NM, 17-Feb-2013.)
 |-  ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex2g 4242* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y (
 <. A ,  B >.  = 
 <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex4g 4243* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( x  =  A  /\  y  =  B )  /\  (
 z  =  C  /\  w  =  D )
 )  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S )  /\  ( C  e.  R  /\  D  e.  S )
 )  ->  ( E. x E. y E. z E. w ( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ps ) )
 
Theorem0nelop 4244 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  <. A ,  B >.
 
Theoremopeqex 4245 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
 |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( C  e.  _V 
 /\  D  e.  _V ) ) )
 
Theoremopcom 4246 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. B ,  A >.  <->  A  =  B )
 
Theoremmoop2 4247* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  B  e.  _V   =>    |-  E* x  A  =  <. B ,  x >.
 
Theoremopeqsn 4248 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
 
Theoremopeqpr 4249 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  { C ,  D }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
 
Theoremeuotd 4250* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( ps 
 <->  ( a  =  A  /\  b  =  B  /\  c  =  C ) ) )   =>    |-  ( ph  ->  E! x E. a E. b E. c ( x  =  <. a ,  b ,  c >.  /\  ps )
 )
 
Theoremuniop 4251 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. <. A ,  B >.  =  { A ,  B }
 
Theoremuniopel 4252 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C )
 
2.3.5  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4253 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( <. x ,  y >.  e.  { <. x ,  y >.  |  ph }  <->  ph )
 
Theoremelopab 4254* Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  { <. x ,  y >.  | 
 ph }  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 )
 
TheoremopelopabsbALT 4255* The law of concretion in terms of substitutions. Less general than opelopabsb 4256, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] [ z  /  x ] ph )
 
Theoremopelopabsb 4256* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
 |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theorembrabsb 4257* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
 |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theoremopelopabt 4258* Closed theorem form of opelopab 4267. (Contributed by NM, 19-Feb-2013.)
 |-  ( ( A. x A. y ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremopelopabga 4259* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ps ) )
 
Theorembrabga 4260* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
 
Theoremopelopab2a 4261* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ps ) )
 
Theoremopelopaba 4262* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theorembraba 4263* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B 
 <->  ps )
 
Theoremopelopabg 4264* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theorembrabg 4265* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
 
Theoremopelopab2 4266* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ch ) )
 
Theoremopelopab 4267* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theorembrab 4268* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  | 
 ph }   =>    |-  ( A R B  <->  ch )
 
Theoremopelopabaf 4269* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4267 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theoremopelopabf 4270* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4267 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.)
 |- 
 F/ x ps   &    |-  F/ y ch   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theoremssopab2 4271 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
 |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps } )
 
Theoremssopab2b 4272 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  ->  ps )
 )
 
Theoremssopab2i 4273 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4274* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
 
Theoremeqopab2b 4275 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  <->  ps ) )
 
Theoremopabm 4276* Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
 |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
 
Theoremiunopab 4277* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
 
2.3.6  Power class of union and intersection
 
Theorempwin 4278 The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |- 
 ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
 
Theorempwunss 4279 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |-  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B )
 
Theorempwssunim 4280 The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
 
Theorempwundifss 4281 Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B )
 
Theorempwunim 4282 The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
 
2.3.7  Epsilon and identity relations
 
Syntaxcep 4283 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4284 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4285* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is,  ( A  _E  B  <->  A  e.  B ) when  B is a set by epelg 4286. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4286 The epsilon relation and membership are the same. General version of epel 4288. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4287 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A  _E  B 
 <->  A  e.  B )
 
Theoremepel 4288 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4289* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5  _I 5 and  -. 4  _I 5. (Contributed by NM, 13-Aug-1995.)
 |- 
 _I  =  { <. x ,  y >.  |  x  =  y }
 
2.3.8  Partial and total orderings

We have not yet defined relations (df-rel 4629), but here we introduce a few related notions we will use to develop ordinals. The class variable  R is no different from other class variables, but it reminds us that typically it represents what we will later call a "relation".

 
Syntaxwpo 4290 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4291 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4292* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression  R  Po  A means  R is a partial order on  A. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Definitiondf-iso 4293* Define the strict linear order predicate. The expression  R  Or  A is true if relationship  R orders  A. The property  x R y  ->  ( x R z  \/  z R y ) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 
x R y  \/  x  =  y  \/  y R x. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) ) ) )
 
Theoremposs 4294 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( A  C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
 
Theorempoeq1 4295 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4296 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4297 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Po  A
 
Theoremnfso 4298 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Or  A
 
Theorempocl 4299 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  Po  A  ->  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) 
 ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
 
Theoremispod 4300* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  -.  x R x )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )   =>    |-  ( ph  ->  R  Po  A )
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