Type  Label  Description 
Statement 

Theorem  ifan 3701 
Rewrite a conjunction in an if statement as two nested conditionals.
(Contributed by Mario Carneiro, 28Jul2014.)



Theorem  ifor 3702 
Rewrite a disjunction in an if statement as two nested conditionals.
(Contributed by Mario Carneiro, 28Jul2014.)



Theorem  dedth 3703 
Weak deduction theorem that eliminates a hypothesis , making it
become an antecedent. We assume that a proof exists for when the
class variable is replaced with a specific class . The
hypothesis
should be assigned to the inference, and the
inference's hypothesis eliminated with elimhyp 3710. If the inference has
other hypotheses with class variable , these can be kept by
assigning keephyp 3716 to them. For more information, see the
Deduction
Theorem http://us.metamath.org/mpeuni/mmdeduction.html.
(Contributed
by NM, 15May1999.)



Theorem  dedth2h 3704 
Weak deduction theorem eliminating two hypotheses. This theorem is
simpler to use than dedth2v 3707 but requires that each hypothesis has
exactly one class variable. See also comments in dedth 3703.
(Contributed by NM, 15May1999.)



Theorem  dedth3h 3705 
Weak deduction theorem eliminating three hypotheses. See comments in
dedth2h 3704. (Contributed by NM, 15May1999.)



Theorem  dedth4h 3706 
Weak deduction theorem eliminating four hypotheses. See comments in
dedth2h 3704. (Contributed by NM, 16May1999.)



Theorem  dedth2v 3707 
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 3704 is simpler to
use. See also comments in dedth 3703. (Contributed by NM, 13Aug1999.)
(Proof shortened by Eric Schmidt, 28Jul2009.)



Theorem  dedth3v 3708 
Weak deduction theorem for eliminating a hypothesis with 3 class
variables. See comments in dedth2v 3707. (Contributed by NM,
13Aug1999.) (Proof shortened by Eric Schmidt, 28Jul2009.)



Theorem  dedth4v 3709 
Weak deduction theorem for eliminating a hypothesis with 4 class
variables. See comments in dedth2v 3707. (Contributed by NM,
21Apr2007.) (Proof shortened by Eric Schmidt, 28Jul2009.)



Theorem  elimhyp 3710 
Eliminate a hypothesis containing class variable when it is known
for a specific class . For more information, see comments in
dedth 3703. (Contributed by NM, 15May1999.)



Theorem  elimhyp2v 3711 
Eliminate a hypothesis containing 2 class variables. (Contributed by
NM, 14Aug1999.)



Theorem  elimhyp3v 3712 
Eliminate a hypothesis containing 3 class variables. (Contributed by
NM, 14Aug1999.)



Theorem  elimhyp4v 3713 
Eliminate a hypothesis containing 4 class variables (for use with the
weak deduction theorem dedth 3703). (Contributed by NM, 16Apr2005.)



Theorem  elimel 3714 
Eliminate a membership hypothesis for weak deduction theorem, when
special case
is provable. (Contributed by NM,
15May1999.)



Theorem  elimdhyp 3715 
Version of elimhyp 3710 where the hypothesis is deduced from the
final
antecedent. See ghomgrplem in set.mm for an example of its use.
(Contributed by Paul Chapman, 25Mar2008.)



Theorem  keephyp 3716 
Transform a hypothesis that we want to keep (but contains the
same class variable used in the eliminated hypothesis) for use
with the weak deduction theorem. (Contributed by NM, 15May1999.)



Theorem  keephyp2v 3717 
Keep a hypothesis containing 2 class variables (for use with the weak
deduction theorem dedth 3703). (Contributed by NM, 16Apr2005.)



Theorem  keephyp3v 3718 
Keep a hypothesis containing 3 class variables. (Contributed by NM,
27Sep1999.)



Theorem  keepel 3719 
Keep a membership hypothesis for weak deduction theorem, when special
case is provable. (Contributed by NM,
14Aug1999.)



Theorem  ifex 3720 
Conditional operator existence. (Contributed by NM, 2Sep2004.)



Theorem  ifexg 3721 
Conditional operator existence. (Contributed by NM, 21Mar2011.)



2.1.15 Power classes


Syntax  cpw 3722 
Extend class notation to include power class. (The tilde in the Metamath
token is meant to suggest the calligraphic font of the P.)



Theorem  pwjust 3723* 
Soundness justification theorem for dfpw 3724. (Contributed by Rodolfo
Medina, 28Apr2010.) (Proof shortened by Andrew Salmon,
29Jun2011.)



Definition  dfpw 3724* 
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
. 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 3 , 5 , 7 , then
3 5 7 3 , 5
3 , 7 5 , 7 3 , 5 , 7 (expw in
set.mm). We will later introduce the Axiom of Power Sets axpow in
set.mm, which can be expressed in class notation per pwexg 4328. Still
later we will prove, in hashpw in set.mm, 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, 5Aug1993.)



Theorem  pweq 3725 
Equality theorem for power class. (Contributed by NM, 5Aug1993.)



Theorem  pweqi 3726 
Equality inference for power class. (Contributed by NM,
27Nov2013.)



Theorem  pweqd 3727 
Equality deduction for power class. (Contributed by NM,
27Nov2013.)



Theorem  elpw 3728 
Membership in a power class. Theorem 86 of [Suppes] p. 47.
(Contributed by NM, 31Dec1993.)



Theorem  elpwg 3729 
Membership in a power class. Theorem 86 of [Suppes] p. 47. See also
elpw2g in set.mm. (Contributed by NM, 6Aug2000.)



Theorem  elpwi 3730 
Subset relation implied by membership in a power class. (Contributed by
NM, 17Feb2007.)



Theorem  elpwid 3731 
An element of a power class is a subclass. Deduction form of elpwi 3730.
(Contributed by David Moews, 1May2017.)



Theorem  elelpwi 3732 
If belongs to a part
of then belongs to .
(Contributed by FL, 3Aug2009.)



Theorem  nfpw 3733 
Boundvariable hypothesis builder for power class. (Contributed by NM,
28Oct2003.) (Revised by Mario Carneiro, 13Oct2016.)



Theorem  pwidg 3734 
Membership of the original in a power set. (Contributed by Stefan O'Rear,
1Feb2015.)



Theorem  pwid 3735 
A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
(Contributed by NM, 5Aug1993.)



Theorem  pwss 3736* 
Subclass relationship for power class. (Contributed by NM,
21Jun2009.)



2.1.16 Unordered and ordered pairs


Syntax  csn 3737 
Extend class notation to include singleton.



Syntax  cpr 3738 
Extend class notation to include unordered pair.



Syntax  ctp 3739 
Extend class notation to include unordered triplet.



Theorem  snjust 3740* 
Soundness justification theorem for dfsn 3741. (Contributed by Rodolfo
Medina, 28Apr2010.) (Proof shortened by Andrew Salmon,
29Jun2011.)



Definition  dfsn 3741* 
Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For
convenience, it is welldefined for proper classes, i.e., those that are
not elements of , although it is not very meaningful in this
case. For an alternate definition see dfsn2 3747. (Contributed by NM,
5Aug1993.)



Definition  dfpr 3742 
Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For
example, 1 , u 1 ^ 2 1 (expr in
set.mm). They are unordered, so as proven by
prcom 3798. For a more traditional definition, but
requiring a dummy
variable, see dfpr2 3749. (Contributed by NM, 5Aug1993.)



Definition  dftp 3743 
Define unordered triple of classes. Definition of [Enderton] p. 19.
(Contributed by NM, 9Apr1994.)



Theorem  sneq 3744 
Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring]
p. 15. (Contributed by NM, 5Aug1993.)



Theorem  sneqi 3745 
Equality inference for singletons. (Contributed by NM, 22Jan2004.)



Theorem  sneqd 3746 
Equality deduction for singletons. (Contributed by NM, 22Jan2004.)



Theorem  dfsn2 3747 
Alternate definition of singleton. Definition 5.1 of [TakeutiZaring]
p. 15. (Contributed by NM, 24Apr1994.)



Theorem  elsn 3748* 
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. (Contributed by NM, 5Aug1993.)



Theorem  dfpr2 3749* 
Alternate definition of unordered pair. Definition 5.1 of
[TakeutiZaring] p. 15.
(Contributed by NM, 24Apr1994.)



Theorem  elprg 3750 
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,
13Sep1995.)



Theorem  elpr 3751 
A member of an unordered pair of classes is one or the other of them.
Exercise 1 of [TakeutiZaring] p.
15. (Contributed by NM,
13Sep1995.)



Theorem  elpr2 3752 
A member of an unordered pair of classes is one or the other of them.
Exercise 1 of [TakeutiZaring] p.
15. (Contributed by NM,
14Oct2005.)



Theorem  elpri 3753 
If a class is an element of a pair, then it is one of the two paired
elements. (Contributed by Scott Fenton, 1Apr2011.)



Theorem  nelpri 3754 
If an element doesn't match the items in an unordered pair, it is not in
the unordered pair. (Contributed by David A. Wheeler, 10May2015.)



Theorem  elsncg 3755 
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15 (generalized). (Contributed by NM, 13Sep1995.) (Proof
shortened by Andrew Salmon, 29Jun2011.)



Theorem  elsnc 3756 
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. (Contributed by NM, 13Sep1995.)



Theorem  elsni 3757 
There is only one element in a singleton. (Contributed by NM,
5Jun1994.)



Theorem  snidg 3758 
A set is a member of its singleton. Part of Theorem 7.6 of [Quine]
p. 49. (Contributed by NM, 28Oct2003.)



Theorem  snidb 3759 
A class is a set iff it is a member of its singleton. (Contributed by NM,
5Apr2004.)



Theorem  snid 3760 
A set is a member of its singleton. Part of Theorem 7.6 of [Quine]
p. 49. (Contributed by NM, 31Dec1993.)



Theorem  elsnc2g 3761 
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. This variation requires only that , rather than , be a
set. (Contributed by NM, 28Oct2003.)



Theorem  elsnc2 3762 
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. This variation requires only that , rather than , be
a set. (Contributed by NM, 12Jun1994.)



Theorem  ralsns 3763* 
Substitution expressed in terms of quantification over a singleton.
(Contributed by Mario Carneiro, 23Apr2015.)



Theorem  rexsns 3764* 
Restricted existential quantification over a singleton. (Contributed by
Mario Carneiro, 23Apr2015.)



Theorem  ralsng 3765* 
Substitution expressed in terms of quantification over a singleton.
(Contributed by NM, 14Dec2005.) (Revised by Mario Carneiro,
23Apr2015.)



Theorem  rexsng 3766* 
Restricted existential quantification over a singleton. (Contributed by
NM, 29Jan2012.)



Theorem  ralsn 3767* 
Convert a quantification over a singleton to a substitution.
(Contributed by NM, 27Apr2009.)



Theorem  rexsn 3768* 
Restricted existential quantification over a singleton. (Contributed by
Jeff Madsen, 5Jan2011.)



Theorem  eltpg 3769 
Members of an unordered triple of classes. (Contributed by FL,
2Feb2014.) (Proof shortened by Mario Carneiro, 11Feb2015.)



Theorem  eltpi 3770 
A member of an unordered triple of classes is one of them. (Contributed
by Mario Carneiro, 11Feb2015.)



Theorem  eltp 3771 
A member of an unordered triple of classes is one of them. Special case
of Exercise 1 of [TakeutiZaring]
p. 17. (Contributed by NM,
8Apr1994.) (Revised by Mario Carneiro, 11Feb2015.)



Theorem  dftp2 3772* 
Alternate definition of unordered triple of classes. Special case of
Definition 5.3 of [TakeutiZaring]
p. 16. (Contributed by NM,
8Apr1994.)



Theorem  nfpr 3773 
Boundvariable hypothesis builder for unordered pairs. (Contributed by
NM, 14Nov1995.)



Theorem  ifpr 3774 
Membership of a conditional operator in an unordered pair. (Contributed
by NM, 17Jun2007.)



Theorem  ralprg 3775* 
Convert a quantification over a pair to a conjunction. (Contributed by
NM, 17Sep2011.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  rexprg 3776* 
Convert a quantification over a pair to a disjunction. (Contributed by
NM, 17Sep2011.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  raltpg 3777* 
Convert a quantification over a triple to a conjunction. (Contributed
by NM, 17Sep2011.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  rextpg 3778* 
Convert a quantification over a triple to a disjunction. (Contributed
by Mario Carneiro, 23Apr2015.)



Theorem  ralpr 3779* 
Convert a quantification over a pair to a conjunction. (Contributed by
NM, 3Jun2007.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  rexpr 3780* 
Convert an existential quantification over a pair to a disjunction.
(Contributed by NM, 3Jun2007.) (Revised by Mario Carneiro,
23Apr2015.)



Theorem  raltp 3781* 
Convert a quantification over a triple to a conjunction. (Contributed
by NM, 13Sep2011.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  rextp 3782* 
Convert a quantification over a triple to a disjunction. (Contributed
by Mario Carneiro, 23Apr2015.)



Theorem  sbcsng 3783* 
Substitution expressed in terms of quantification over a singleton.
(Contributed by NM, 14Dec2005.) (Revised by Mario Carneiro,
23Apr2015.)



Theorem  nfsn 3784 
Boundvariable hypothesis builder for singletons. (Contributed by NM,
14Nov1995.)



Theorem  csbsng 3785 
Distribute proper substitution through the singleton of a class.
csbsng 3785 is derived from the virtual deduction proof
csbsngVD in
set.mm. (Contributed by Alan Sare, 10Nov2012.)



Theorem  disjsn 3786 
Intersection with the singleton of a nonmember is disjoint.
(Contributed by NM, 22May1998.) (Proof shortened by Andrew Salmon,
29Jun2011.) (Proof shortened by Wolf Lammen, 30Sep2014.)



Theorem  disjsn2 3787 
Intersection of distinct singletons is disjoint. (Contributed by NM,
25May1998.)



Theorem  snprc 3788 
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, 5Aug1993.)



Theorem  r19.12sn 3789* 
Special case of r19.12 2727 where its converse holds. (Contributed by
NM,
19May2008.) (Revised by Mario Carneiro, 23Apr2015.)



Theorem  rabsn 3790* 
Condition where a restricted class abstraction is a singleton.
(Contributed by NM, 28May2006.)



Theorem  euabsn2 3791* 
Another way to express existential uniqueness of a wff: its class
abstraction is a singleton. (Contributed by Mario Carneiro,
14Nov2016.)



Theorem  euabsn 3792 
Another way to express existential uniqueness of a wff: its class
abstraction is a singleton. (Contributed by NM, 22Feb2004.)



Theorem  reusn 3793* 
A way to express restricted existential uniqueness of a wff: its
restricted class abstraction is a singleton. (Contributed by NM,
30May2006.) (Proof shortened by Mario Carneiro, 14Nov2016.)



Theorem  absneu 3794 
Restricted existential uniqueness determined by a singleton.
(Contributed by NM, 29May2006.)



Theorem  rabsneu 3795 
Restricted existential uniqueness determined by a singleton.
(Contributed by NM, 29May2006.) (Revised by Mario Carneiro,
23Dec2016.)



Theorem  eusn 3796* 
Two ways to express " is a singleton." (Contributed by NM,
30Oct2010.)



Theorem  rabsnt 3797* 
Truth implied by equality of a restricted class abstraction and a
singleton. (Contributed by NM, 29May2006.) (Proof shortened by Mario
Carneiro, 23Dec2016.)



Theorem  prcom 3798 
Commutative law for unordered pairs. (Contributed by NM, 5Aug1993.)



Theorem  preq1 3799 
Equality theorem for unordered pairs. (Contributed by NM,
29Mar1998.)



Theorem  preq2 3800 
Equality theorem for unordered pairs. (Contributed by NM, 5Aug1993.)

