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Type | Label | Description |
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Statement | ||
Theorem | difdifdirss 3301 | Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Theorem | uneqdifeqim 3302 | Two ways that and can "partition" (when and don't overlap and is a part of ). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Theorem | r19.2m 3303* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1526). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | r19.3rm 3304* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | r19.28m 3305* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | r19.3rmv 3306* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | r19.9rmv 3307* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | r19.28mv 3308* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | r19.45mv 3309* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Theorem | r19.27m 3310* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | r19.27mv 3311* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | rzal 3312* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | rexn0 3313* | Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Theorem | rexm 3314* | Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) |
Theorem | ralidm 3315* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Theorem | ral0 3316 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
Theorem | rgenm 3317* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | ralf0 3318* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
Theorem | ralm 3319 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaanlem 3320* | Special case of raaan 3321 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaan 3321* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
Theorem | raaanv 3322* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Theorem | sbss 3323* | 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 3324 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
Syntax | cif 3325 | Extend class notation to include the conditional operator. See df-if 3326 for a description. (In older databases this was denoted "ded".) |
Definition | df-if 3326* |
Define the conditional operator. Read as "if
then
else ." See iftrue 3330 and iffalse 3333 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 742). (Contributed by NM, 15-May-1999.) |
Theorem | dfif6 3327* | An alternate definition of the conditional operator df-if 3326 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq1 3328 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq2 3329 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | iftrue 3330 | 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 3331 | Inference associated with iftrue 3330. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iftrued 3332 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | iffalse 3333 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
Theorem | iffalsei 3334 | Inference associated with iffalse 3333. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iffalsed 3335 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | ifnefalse 3336 | 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 3333 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
Theorem | dfif3 3337* | Alternate definition of the conditional operator df-if 3326. 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 | ifeq12 3338 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
Theorem | ifeq1d 3339 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq2d 3340 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq12d 3341 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Theorem | ifbi 3342 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Theorem | ifbid 3343 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Theorem | ifbieq1d 3344 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
Theorem | ifbieq2i 3345 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq2d 3346 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3347 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3348 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3349 | Deduction version of nfif 3350. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3350 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Syntax | cpw 3351 | 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 3352* | Soundness justification theorem for df-pw 3353. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-pw 3353* | 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 3354 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3355 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3356 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3357 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | selpw 3358* | Setvar variable membership in a power class (common case). See elpw 3357. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3359 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3360 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwid 3361 | An element of a power class is a subclass. Deduction form of elpwi 3360. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3362 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3363 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3364 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3365 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3366* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3367 | Extend class notation to include singleton. |
Syntax | cpr 3368 | Extend class notation to include unordered pair. |
Syntax | ctp 3369 | Extend class notation to include unordered triplet. |
Syntax | cop 3370 | Extend class notation to include ordered pair. |
Syntax | cotp 3371 | Extend class notation to include ordered triple. |
Theorem | snjust 3372* | Soundness justification theorem for df-sn 3373. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-sn 3373* | 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 3381. (Contributed by NM, 5-Aug-1993.) |
Definition | df-pr 3374 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3437. For a more traditional definition, but requiring a dummy variable, see dfpr2 3383. (Contributed by NM, 5-Aug-1993.) |
Definition | df-tp 3375 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-op 3376* |
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 3562 and opprc2 3563). For
Kuratowski's actual definition when the arguments are sets, see dfop 3539.
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3376 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3376 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 3377 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3378 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3379 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3380 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3381 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsn 3382* | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | dfpr2 3383* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 3384 | 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 3385 | 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 3386 | 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 3387 | 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 3388 | 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 | elsncg 3389 | 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.) |
Theorem | elsnc 3390 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | elsni 3391 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | snidg 3392 | 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 3393 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Theorem | snid 3394 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.) |
Theorem | ssnid 3395 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elsnc2g 3396 | 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, 28-Oct-2003.) |
Theorem | elsnc2 3397 | 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, 12-Jun-1994.) |
Theorem | mosn 3398* | A singleton has at most one element. This works whether is a proper class or not, and in that sense can be seen as encompassing both snmg 3477 and snprc 3426. (Contributed by Jim Kingdon, 30-Aug-2018.) |
Theorem | ralsns 3399* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsns 3400* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
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