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Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremifboth 3501 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   =>    |-  ( ( ps 
 /\  ch )  ->  th )
 
Theoremifid 3502 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
 |- 
 if ( ph ,  A ,  A )  =  A
 
Theoremeqif 3503 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  =  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) ) )
 
Theoremelif 3504 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  e.  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C ) ) )
 
Theoremifel 3505 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
 |-  ( if ( ph ,  A ,  B )  e.  C  <->  ( ( ph  /\  A  e.  C )  \/  ( -.  ph  /\  B  e.  C ) ) )
 
Theoremifcl 3506 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  if ( ph ,  A ,  B )  e.  C )
 
Theoremifeqor 3507 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
 
Theoremifnot 3508 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
 |- 
 if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
 
Theoremifan 3509 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  /\ 
 ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B )
 
Theoremifor 3510 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  \/  ps ) ,  A ,  B )  =  if ( ph ,  A ,  if ( ps ,  A ,  B ) )
 
Theoremdedth 3511 Weak deduction theorem that eliminates a hypothesis  ph, making it become an antecedent. We assume that a proof exists for  ph when the class variable  A is replaced with a specific class 
B. The hypothesis  ch should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3518. If the inference has other hypotheses with class variable  A, these can be kept by assigning keephyp 3524 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpegif/mmdeduction.html. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ch   =>    |-  ( ph  ->  ps )
 
Theoremdedth2h 3512 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3515 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3511. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ch  <->  th ) )   &    |-  ( B  =  if ( ps ,  B ,  D )  ->  ( th 
 <->  ta ) )   &    |-  ta   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremdedth3h 3513 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3512. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( th  <->  ta ) )   &    |-  ( B  =  if ( ps ,  B ,  R )  ->  ( ta 
 <->  et ) )   &    |-  ( C  =  if ( ch ,  C ,  S )  ->  ( et  <->  ze ) )   &    |-  ze   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
 
Theoremdedth4h 3514 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3512. (Contributed by NM, 16-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ta  <->  et ) )   &    |-  ( B  =  if ( ps ,  B ,  S )  ->  ( et 
 <->  ze ) )   &    |-  ( C  =  if ( ch ,  C ,  F )  ->  ( ze  <->  si ) )   &    |-  ( D  =  if ( th ,  D ,  G )  ->  ( si  <->  rh ) )   &    |-  rh   =>    |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
 
Theoremdedth2v 3515 Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3512 is simpler to use. See also comments in dedth 3511. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  th   =>    |-  ( ph  ->  ps )
 
Theoremdedth3v 3516 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3515. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ta   =>    |-  ( ph  ->  ps )
 
Theoremdedth4v 3517 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3515. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  S )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  T )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  D ,  U )  ->  ( ta  <->  et ) )   &    |-  et   =>    |-  ( ph  ->  ps )
 
Theoremelimhyp 3518 Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3511. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <->  ps ) )   &    |-  ch   =>    |-  ps
 
Theoremelimhyp2v 3519 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ta   =>    |-  th
 
Theoremelimhyp3v 3520 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  et   =>    |-  ta
 
Theoremelimhyp4v 3521 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3511). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh ) )   &    |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps ) )   &    |-  et   =>    |-  ps
 
Theoremelimel 3522 Eliminate a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 15-May-1999.)
 |-  B  e.  C   =>    |-  if ( A  e.  C ,  A ,  B )  e.  C
 
Theoremelimdhyp 3523 Version of elimhyp 3518 where the hypothesis is deduced from the final antecedent. See ghomgrplem 23167 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
 |-  ( ph  ->  ps )   &    |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch ) )   &    |-  th   =>    |-  ch
 
Theoremkeephyp 3524 Transform a hypothesis  ps that we want to keep (but contains the same class variable  A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  ps   &    |-  ch   =>    |-  th
 
Theoremkeephyp2v 3525 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3511). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ps   &    |-  ta   =>    |-  th
 
Theoremkeephyp3v 3526 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( rh  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  rh   &    |-  et   =>    |-  ta
 
Theoremkeepel 3527 Keep a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 14-Aug-1999.)
 |-  A  e.  C   &    |-  B  e.  C   =>    |- 
 if ( ph ,  A ,  B )  e.  C
 
Theoremifex 3528 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 if ( ph ,  A ,  B )  e.  _V
 
Theoremifexg 3529 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
 
2.1.16  Power classes
 
Syntaxcpw 3530 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
 class  ~P A
 
Theorempwjust 3531* Soundness justification theorem for df-pw 3532. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  C_  A }  =  {
 y  |  y  C_  A }
 
Definitiondf-pw 3532* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of  _V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if  A  =  { 3 ,  5 ,  7 }, then  ~P A  =  { (/) ,  { 3 } ,  { 5 } ,  { 7 } ,  { 3 ,  5 } ,  { 3 ,  7 } ,  {
5 ,  7 } ,  { 3 ,  5 ,  7 } } (ex-pw 20629). We will later introduce the Axiom of Power Sets ax-pow 4082, which can be expressed in class notation per pwexg 4088. Still later we will prove, in hashpw 11265, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
 |- 
 ~P A  =  { x  |  x  C_  A }
 
Theorempweq 3533 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 3534 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 3535 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 3536 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  ~P B  <->  A  C_  B )
 
Theoremelpwg 3537 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4063. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 3538 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelelpwi 3539 If  A belongs to a part of  C then  A belongs to  C. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  e.  B  /\  B  e.  ~P C )  ->  A  e.  C )
 
Theoremnfpw 3540 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ~P A
 
Theorempwidg 3541 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
 
Theorempwid 3542 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  ~P A
 
Theorempwss 3543* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
 |-  ( ~P A  C_  B 
 <-> 
 A. x ( x 
 C_  A  ->  x  e.  B ) )
 
2.1.17  Unordered and ordered pairs
 
Syntaxcsn 3544 Extend class notation to include singleton.
 class  { A }
 
Syntaxcpr 3545 Extend class notation to include unordered pair.
 class  { A ,  B }
 
Syntaxctp 3546 Extend class notation to include unordered triplet.
 class  { A ,  B ,  C }
 
Syntaxcop 3547 Extend class notation to include ordered pair.
 class  <. A ,  B >.
 
Syntaxcotp 3548 Extend class notation to include ordered triple.
 class  <. A ,  B ,  C >.
 
Theoremsnjust 3549* Soundness justification theorem for df-sn 3550. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  =  A }  =  {
 y  |  y  =  A }
 
Definitiondf-sn 3550* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of  _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3558. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  =  { x  |  x  =  A }
 
Definitiondf-pr 3551 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example,  A  e.  {
1 ,  -u 1 }  ->  ( A ^
2 )  =  1 (ex-pr 20630). They are unordered, so  { A ,  B }  =  { B ,  A } as proven by prcom 3609. For a more traditional definition, but requiring a dummy variable, see dfpr2 3560. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  ( { A }  u.  { B } )
 
Definitiondf-tp 3552 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  ( { A ,  B }  u.  { C }
 )
 
Definitiondf-op 3553* Definition of an ordered pair, equivalent to Kuratowski's definition  { { A } ,  { A ,  B } } when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3718, opprc2 3719, and 0nelop 4149). For Kuratowski's actual definition when the arguments are sets, see dfop 3695. For the justifying theorem (for sets) see opth 4138. See dfopif 3693 for an equivalent formulation using the  if operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as  <. A ,  B >.  =  { { A } ,  { A ,  B } }, which has different behavior from our df-op 3553 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3553 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition  <. A ,  B >._2  =  { { { A } ,  (/) } ,  { { B } } }, justified by opthwiener 4161. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition  <. A ,  B >._3  =  { A ,  { A ,  B } } is justified by opthreg 7203, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is  <. A ,  B >._4  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) ), justified by opthprc 4643. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 11163. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9618. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

 |- 
 <. A ,  B >.  =  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
 
Definitiondf-ot 3554 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
 |- 
 <. A ,  B ,  C >.  =  <. <. A ,  B >. ,  C >.
 
Theoremsneq 3555 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { A }  =  { B } )
 
Theoremsneqi 3556 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  A  =  B   =>    |-  { A }  =  { B }
 
Theoremsneqd 3557 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A }  =  { B } )
 
Theoremdfsn2 3558 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A }  =  { A ,  A }
 
Theoremelsn 3559* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  { A }  <->  x  =  A )
 
Theoremdfpr2 3560* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A ,  B }  =  { x  |  ( x  =  A  \/  x  =  B ) }
 
Theoremelprg 3561 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
 |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 ) )
 
Theoremelpr 3562 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
 |-  A  e.  _V   =>    |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 )
 
Theoremelpr2 3563 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 )
 
Theoremelpri 3564 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
 |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
 
Theoremnelpri 3565 If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |- 
 -.  A  e.  { B ,  C }
 
Theoremelsncg 3566 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( A  e.  { B }  <->  A  =  B ) )
 
Theoremelsnc 3567 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
 |-  A  e.  _V   =>    |-  ( A  e.  { B }  <->  A  =  B )
 
Theoremelsni 3568 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  { B }  ->  A  =  B )
 
Theoremsnidg 3569 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
 |-  ( A  e.  V  ->  A  e.  { A } )
 
Theoremsnidb 3570 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  e.  _V  <->  A  e.  { A } )
 
Theoremsnid 3571 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A }
 
Theoremelsnc2g 3572 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 28-Oct-2003.)
 |-  ( B  e.  V  ->  ( A  e.  { B }  <->  A  =  B ) )
 
Theoremelsnc2 3573 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 12-Jun-1994.)
 |-  B  e.  _V   =>    |-  ( A  e.  { B }  <->  A  =  B )
 
Theoremralsns 3574* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( A. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremrexsns 3575* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 3576* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
 
Theoremrexsng 3577* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
 
Theoremralsn 3578* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  { A } ph  <->  ps )
 
Theoremrexsn 3579* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  { A } ph  <->  ps )
 
Theoremeltpg 3580 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) ) )
 
Theoremeltpi 3581 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  { B ,  C ,  D }  ->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremeltp 3582 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  A  e.  _V   =>    |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremdftp2 3583* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
 
Theoremnfpr 3584 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremifpr 3585 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
 
Theoremralprg 3586* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\  ch ) ) )
 
Theoremrexprg 3587* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) ) )
 
Theoremraltpg 3588* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  /\  ch  /\ 
 th ) ) )
 
Theoremrextpg 3589* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  \/  ch  \/  th ) ) )
 
Theoremralpr 3590* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( A. x  e. 
 { A ,  B } ph  <->  ( ps  /\  ch ) )
 
Theoremrexpr 3591* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( E. x  e. 
 { A ,  B } ph  <->  ( ps  \/  ch ) )
 
Theoremraltp 3592* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( A. x  e.  { A ,  B ,  C } ph  <->  ( ps  /\  ch 
 /\  th ) )
 
Theoremrextp 3593* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( E. x  e.  { A ,  B ,  C } ph  <->  ( ps  \/  ch 
 \/  th ) )
 
Theoremsbcsng 3594* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x  e.  { A } ph ) )
 
Theoremnfsn 3595 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 3596 Distribute proper substitution through the singleton of a class. csbsng 3596 is derived from the virtual deduction proof csbsngVD 27359. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremdisjsn 3597 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
 |-  ( ( A  i^i  { B } )  =  (/) 
 <->  -.  B  e.  A )
 
Theoremdisjsn2 3598 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremsnprc 3599 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
 
Theoremr19.12sn 3600* Special case of r19.12 2618 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph )
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