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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rzal 3501* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | rexn0 3502* | Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3503). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Theorem | rexm 3503* | Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) |
Theorem | ralidm 3504* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Theorem | ral0 3505 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
Theorem | rgenm 3506* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | ralf0 3507* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
Theorem | ralm 3508 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaanlem 3509* | Special case of raaan 3510 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaan 3510* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
Theorem | raaanv 3511* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Theorem | sbss 3512* | Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Theorem | sbcssg 3513 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
Theorem | dcun 3514 | The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) |
DECID DECID DECID | ||
Syntax | cif 3515 | Extend class notation to include the conditional operator. See df-if 3516 for a description. (In older databases this was denoted "ded".) |
Definition | df-if 3516* |
Define the conditional operator. Read as "if
then
else ." See iftrue 3520 and iffalse 3523 for its
values. In mathematical literature, this operator is rarely defined
formally but is implicit in informal definitions such as "let
f(x)=0 if
x=0 and 1/x otherwise."
In the absence of excluded middle, this will tend to be useful where is decidable (in the sense of df-dc 825). (Contributed by NM, 15-May-1999.) |
Theorem | dfif6 3517* | An alternate definition of the conditional operator df-if 3516 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq1 3518 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq2 3519 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | iftrue 3520 | 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.) |
Theorem | iftruei 3521 | Inference associated with iftrue 3520. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iftrued 3522 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | iffalse 3523 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
Theorem | iffalsei 3524 | Inference associated with iffalse 3523. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iffalsed 3525 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | ifnefalse 3526 | 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 versus applying iffalse 3523 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
Theorem | ifsbdc 3527 | Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | dfif3 3528* | Alternate definition of the conditional operator df-if 3516. 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.) |
Theorem | ifssun 3529 | A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
Theorem | ifidss 3530 | A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
Theorem | ifeq12 3531 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
Theorem | ifeq1d 3532 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq2d 3533 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq12d 3534 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Theorem | ifbi 3535 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Theorem | ifbid 3536 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Theorem | ifbieq1d 3537 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
Theorem | ifbieq2i 3538 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq2d 3539 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3540 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3541 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3542 | Deduction version of nfif 3543. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3543 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifcldadc 3544 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
DECID | ||
Theorem | ifeq1dadc 3545 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | ifbothdadc 3546 | A formula containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.) |
DECID | ||
Theorem | ifbothdc 3547 | A wff containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifiddc 3548 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
DECID | ||
Theorem | eqifdc 3549 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
DECID | ||
Theorem | ifcldcd 3550 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifandc 3551 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
DECID | ||
Theorem | ifmdc 3552 | If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.) |
DECID | ||
Syntax | cpw 3553 | 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 3554* | Soundness justification theorem for df-pw 3555. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-pw 3555* | 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 is { 3 , 5 , 7 }, then is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove 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.) |
Theorem | pweq 3556 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3557 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3558 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3559 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | velpw 3560* | Setvar variable membership in a power class (common case). See elpw 3559. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3561 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3562 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwb 3563 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Theorem | elpwid 3564 | An element of a power class is a subclass. Deduction form of elpwi 3562. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3565 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3566 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3567 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3568 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3569* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3570 | Extend class notation to include singleton. |
Syntax | cpr 3571 | Extend class notation to include unordered pair. |
Syntax | ctp 3572 | Extend class notation to include unordered triplet. |
Syntax | cop 3573 | Extend class notation to include ordered pair. |
Syntax | cotp 3574 | Extend class notation to include ordered triple. |
Theorem | snjust 3575* | Soundness justification theorem for df-sn 3576. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-sn 3576* | 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 3584. (Contributed by NM, 5-Aug-1993.) |
Definition | df-pr 3577 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3646. For a more traditional definition, but requiring a dummy variable, see dfpr2 3589. (Contributed by NM, 5-Aug-1993.) |
Definition | df-tp 3578 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-op 3579* |
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 3774 and opprc2 3775). For
Kuratowski's actual definition when the arguments are sets, see dfop 3751.
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3579 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3579 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 . This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is 3 , but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Definition | df-ot 3580 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3581 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3582 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3583 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3584 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 3585 | There is exactly 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.) |
Theorem | elsn 3586 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3587 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3588 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3589* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 3590 | 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.) |
Theorem | elpr 3591 | 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.) |
Theorem | elpr2 3592 | 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.) |
Theorem | elpri 3593 | 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.) |
Theorem | nelpri 3594 | 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.) |
Theorem | prneli 3595 | If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.) |
Theorem | nelprd 3596 | If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
Theorem | eldifpr 3597 | Membership in a set with two elements removed. Similar to eldifsn 3697 and eldiftp 3616. (Contributed by Mario Carneiro, 18-Jul-2017.) |
Theorem | rexdifpr 3598 | Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
Theorem | snidg 3599 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.) |
Theorem | snidb 3600 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
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