HomeHome Intuitionistic Logic Explorer
Theorem List (p. 42 of 133)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxpow3 4101* A variant of the Axiom of Power Sets ax-pow 4098. For any set  x, there exists a set  y whose members are exactly the subsets of  x i.e. the power set of  x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 |- 
 E. y A. z
 ( z  C_  x  <->  z  e.  y )
 
Theoremel 4102* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |- 
 E. y  x  e.  y
 
Theoremvpwex 4103 Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4104 from vpwex 4103. (Revised by BJ, 10-Aug-2022.)
 |- 
 ~P x  e.  _V
 
Theorempwexg 4104 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  e.  V  ->  ~P A  e.  _V )
 
Theorempwexd 4105 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ~P A  e.  _V )
 
Theoremabssexg 4106* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  e.  V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
 
Theorempwex 4107 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
 |-  A  e.  _V   =>    |-  ~P A  e.  _V
 
Theoremsnexg 4108 A singleton whose element exists is a set. The  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 |-  ( A  e.  V  ->  { A }  e.  _V )
 
Theoremsnex 4109 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  e.  _V   =>    |-  { A }  e.  _V
 
Theoremsnexprc 4110 A singleton whose element is a proper class is a set. The  -.  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 |-  ( -.  A  e.  _V 
 ->  { A }  e.  _V )
 
Theoremnotnotsnex 4111 A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
 |- 
 -.  -.  { A }  e.  _V
 
Theoremp0ex 4112 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
 |- 
 { (/) }  e.  _V
 
Theorempp0ex 4113  { (/) ,  { (/)
} } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
 |- 
 { (/) ,  { (/) } }  e.  _V
 
Theoremord3ex 4114 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
 |- 
 { (/) ,  { (/) } ,  { (/) ,  { (/) } } }  e.  _V
 
Theoremdtruarb 4115* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4474 in which we are given a set  y and go from there to a set  x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
 |- 
 E. x E. y  -.  x  =  y
 
Theorempwuni 4116 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
 |-  A  C_  ~P U. A
 
Theoremundifexmid 4117* Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3443 and undifdcss 6811 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
 |-  ( x  C_  y  <->  ( x  u.  ( y 
 \  x ) )  =  y )   =>    |-  ( ph  \/  -.  ph )
 
2.3.2  A notation for excluded middle
 
Syntaxwem 4118 Formula for an abbreviation of excluded middle.
 wff EXMID
 
Definitiondf-exmid 4119 The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions.

To see how this compares with other ways of expressing excluded middle, compare undifexmid 4117 with exmidundif 4129. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show  ph and  -.  ph in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis  ph  \/  -.  ph for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID  ph by exmidexmid 4120 but there is no good way to express the converse.

This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4046, in which case EXMID means that all propositions are decidable (see exmidexmid 4120 and notice that it relies on ax-sep 4046). If we instead work with ax-bdsep 13112, EXMID as defined here means that all bounded propositions are decidable.

(Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.)

 |-  (EXMID  <->  A. x ( x  C_  { (/) }  -> DECID  (/)  e.  x ) )
 
Theoremexmidexmid 4120 EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 828, peircedc 899, or condc 838.

(Contributed by Jim Kingdon, 18-Jun-2022.)

 |-  (EXMID 
 -> DECID  ph )
 
Theoremexmid01 4121 Excluded middle is equivalent to saying any subset of  { (/)
} is either  (/) or  { (/) }. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
 
Theorempwntru 4122 A slight strengthening of pwtrufal 13222. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
 |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  =  (/) )
 
Theoremexmid1dc 4123* A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4117 or ordtriexmid 4437. In this context  x  =  { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  ( ( ph  /\  x  C_ 
 { (/) } )  -> DECID  x  =  { (/) } )   =>    |-  ( ph  -> EXMID )
 
Theoremexmidn0m 4124* Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
 |-  (EXMID  <->  A. x ( x  =  (/)  \/  E. y  y  e.  x ) )
 
Theoremexmidsssn 4125* Excluded middle is equivalent to the biconditionalized version of sssnr 3680 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
 |-  (EXMID  <->  A. x A. y ( x  C_  { y } 
 <->  ( x  =  (/)  \/  x  =  { y } ) ) )
 
Theoremexmidsssnc 4126* Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4121 but lets you choose any set as the element of the singleton rather than just  (/). It is similar to exmidsssn 4125 but for a particular set  B rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
 |-  ( B  e.  V  ->  (EXMID  <->  A. x ( x  C_  { B }  ->  ( x  =  (/)  \/  x  =  { B } )
 ) ) )
 
Theoremexmid0el 4127 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 4128* 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 4129* 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 3443 and undifdcss 6811 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 4130* Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4129 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 4131* 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 4049). (Contributed by NM, 14-Nov-2006.)
 |- 
 E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
 
Theoremzfpair2 4132 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4131. (Contributed by NM, 14-Nov-2006.)
 |- 
 { x ,  y }  e.  _V
 
Theoremprexg 4133 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 3633, prprc1 3631, and prprc2 3632. (Contributed by Jim Kingdon, 16-Sep-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
 
Theoremsnelpwi 4134 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 4135 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 4136 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 4137* 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 4138 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 4139 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 4140 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 4141 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 4142* 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 4143* 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 4144* Negation of subclass relationship. Compare nssr 3157. (Contributed by Jim Kingdon, 17-Sep-2018.)
 |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A  C_  B )
 
Theorempweqb 4145 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 4146* 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 4147 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
 |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
 
Theoremmss 4148* 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 4149* 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 4150 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 4151 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 4152 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 4153 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 4154 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 4155 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 4156* 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 4157 An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4156). (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  =/=  (/)
 
Theoremopth1 4158 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 4159 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 4160 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 4161 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 4162 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremotth2 4163 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 4164 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 4165* 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 4166* 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 4167* Closed theorem form of copsex2g 4168. (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 4168* 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 4169* 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 4170 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  <. A ,  B >.
 
Theoremopeqex 4171 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 4172 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 4173* "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 4174 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 4175 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 4176* 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 4177 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 4178 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 4179 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 4180* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  { <. x ,  y >.  | 
 ph }  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 )
 
TheoremopelopabsbALT 4181* The law of concretion in terms of substitutions. Less general than opelopabsb 4182, 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 4182* 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 4183* 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 4184* Closed theorem form of opelopab 4193. (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 4185* 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 4186* 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 4187* 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 4188* 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 4189* 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 4190* 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 4191* 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 4192* 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 4193* 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 4194* 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 4195* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4193 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 4196* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4193 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 4197 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 4198 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 4199 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4200* 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 } )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13283
  Copyright terms: Public domain < Previous  Next >