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Theorem List for Metamath Proof Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfif2 3701* An alternate definition of the conditional operator df-if 3700 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)

Theoremdfif6 3702* An alternate definition of the conditional operator df-if 3700 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3703 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremifeq2 3704 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremiftrue 3705 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremiffalse 3706 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Theoremifnefalse 3707 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3706 directly in this case. It happens, e.g., in oevn0 6718. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremifsb 3708 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)

Theoremdfif3 3709* Alternate definition of the conditional operator df-if 3700. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremdfif4 3710* Alternate definition of the conditional operator df-if 3700. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Theoremdfif5 3711* Alternate definition of the conditional operator df-if 3700. Note that is independent of i.e. a constant true or false (see also abvor0 3605). (Contributed by Gérard Lang, 18-Aug-2013.)

Theoremifeq12 3712 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)

Theoremifeq1d 3713 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq2d 3714 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq12d 3715 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Theoremifbi 3716 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Theoremifbid 3717 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)

Theoremifbieq2i 3718 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq2d 3719 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq12i 3720 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Theoremifbieq12d 3721 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfifd 3722 Deduction version of nfif 3723. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremnfif 3723 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifeq1da 3724 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifeq2da 3725 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifclda 3726 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremcsbifg 3727 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelimif 3728 Elimination of a conditional operator contained in a wff . (Contributed by NM, 15-Feb-2005.)

Theoremifbothda 3729 A wff containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)

Theoremifboth 3730 A wff 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.)

Theoremifid 3731 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Theoremeqif 3732 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremelif 3733 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremifel 3734 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)

Theoremifcl 3735 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)

Theoremifeqor 3736 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifnot 3737 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)

Theoremifan 3738 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremifor 3739 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremdedth 3740 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 3747. If the inference has other hypotheses with class variable , these can be kept by assigning keephyp 3753 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpeuni/mmdeduction.html. (Contributed by NM, 15-May-1999.)

Theoremdedth2h 3741 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3744 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3740. (Contributed by NM, 15-May-1999.)

Theoremdedth3h 3742 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3741. (Contributed by NM, 15-May-1999.)

Theoremdedth4h 3743 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3741. (Contributed by NM, 16-May-1999.)

Theoremdedth2v 3744 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 3741 is simpler to use. See also comments in dedth 3740. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth3v 3745 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3744. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth4v 3746 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3744. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremelimhyp 3747 Eliminate a hypothesis containing class variable when it is known for a specific class . For more information, see comments in dedth 3740. (Contributed by NM, 15-May-1999.)

Theoremelimhyp2v 3748 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp3v 3749 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp4v 3750 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3740). (Contributed by NM, 16-Apr-2005.)

Theoremelimel 3751 Eliminate a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 15-May-1999.)

Theoremelimdhyp 3752 Version of elimhyp 3747 where the hypothesis is deduced from the final antecedent. See ghomgrplem 25053 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremkeephyp 3753 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, 15-May-1999.)

Theoremkeephyp2v 3754 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3740). (Contributed by NM, 16-Apr-2005.)

Theoremkeephyp3v 3755 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)

Theoremkeepel 3756 Keep a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 14-Aug-1999.)

Theoremifex 3757 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)

Theoremifexg 3758 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)

2.1.16  Power classes

Syntaxcpw 3759 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)

Theorempwjust 3760* Soundness justification theorem for df-pw 3761. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-pw 3761* 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 , then (ex-pw 21690). We will later introduce the Axiom of Power Sets ax-pow 4337, which can be expressed in class notation per pwexg 4343. Still later we will prove, in hashpw 11654, 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.)

Theorempweq 3762 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Theorempweqi 3763 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)

Theorempweqd 3764 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)

Theoremelpw 3765 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Theoremelpwg 3766 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4323. (Contributed by NM, 6-Aug-2000.)

Theoremelpwi 3767 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Theoremelpwid 3768 An element of a power class is a subclass. Deduction form of elpwi 3767. (Contributed by David Moews, 1-May-2017.)

Theoremelelpwi 3769 If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.)

Theoremnfpw 3770 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theorempwidg 3771 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theorempwid 3772 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)

Theorempwss 3773* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

2.1.17  Unordered and ordered pairs

Syntaxcsn 3774 Extend class notation to include singleton.

Syntaxcpr 3775 Extend class notation to include unordered pair.

Syntaxctp 3776 Extend class notation to include unordered triplet.

Syntaxcop 3777 Extend class notation to include ordered pair.

Syntaxcotp 3778 Extend class notation to include ordered triple.

Theoremsnjust 3779* Soundness justification theorem for df-sn 3780. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-sn 3780* 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 , although it is not very meaningful in this case. For an alternate definition see dfsn2 3788. (Contributed by NM, 5-Aug-1993.)

Definitiondf-pr 3781 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, (ex-pr 21691). They are unordered, so as proven by prcom 3842. For a more traditional definition, but requiring a dummy variable, see dfpr2 3790. (Contributed by NM, 5-Aug-1993.)

Definitiondf-tp 3782 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)

Definitiondf-op 3783* Definition of an ordered pair, equivalent to Kuratowski's definition 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 3966, opprc2 3967, and 0nelop 4406). For Kuratowski's actual definition when the arguments are sets, see dfop 3943. For the justifying theorem (for sets) see opth 4395. See dfopif 3941 for an equivalent formulation using the operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3783 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3783 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 _2 , justified by opthwiener 4418. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 7529, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is _4 , justified by opthprc 4884. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 11518. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9948. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Definitiondf-ot 3784 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)

Theoremsneq 3785 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremsneqi 3786 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremsneqd 3787 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremdfsn2 3788 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelsn 3789* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremdfpr2 3790* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelprg 3791 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.)

Theoremelpr 3792 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.)

Theoremelpr2 3793 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.)

Theoremelpri 3794 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.)

Theoremnelpri 3795 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.)

Theoremelsncg 3796 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.)

Theoremelsnc 3797 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremelsni 3798 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)

Theoremsnidg 3799 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Theoremsnidb 3800 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

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