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Type | Label | Description |
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Statement | ||
Theorem | r19.9rmv 3401* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | r19.28mv 3402* | 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 3403* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Theorem | r19.44mv 3404* | Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Theorem | r19.27m 3405* | 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 3406* | 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 3407* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | rexn0 3408* | Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3409). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Theorem | rexm 3409* | Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) |
Theorem | ralidm 3410* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Theorem | ral0 3411 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
Theorem | rgenm 3412* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Theorem | ralf0 3413* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
Theorem | ralm 3414 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaanlem 3415* | Special case of raaan 3416 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Theorem | raaan 3416* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
Theorem | raaanv 3417* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Theorem | sbss 3418* | 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 3419 | 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 3420 | The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) |
DECID DECID DECID | ||
Syntax | cif 3421 | Extend class notation to include the conditional operator. See df-if 3422 for a description. (In older databases this was denoted "ded".) |
Definition | df-if 3422* |
Define the conditional operator. Read as "if
then
else ." See iftrue 3426 and iffalse 3429 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 787). (Contributed by NM, 15-May-1999.) |
Theorem | dfif6 3423* | An alternate definition of the conditional operator df-if 3422 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq1 3424 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq2 3425 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | iftrue 3426 | 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 3427 | Inference associated with iftrue 3426. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iftrued 3428 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | iffalse 3429 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
Theorem | iffalsei 3430 | Inference associated with iffalse 3429. (Contributed by BJ, 7-Oct-2018.) |
Theorem | iffalsed 3431 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | ifnefalse 3432 | 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 3429 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
Theorem | ifsbdc 3433 | Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | dfif3 3434* | Alternate definition of the conditional operator df-if 3422. 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 3435 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
Theorem | ifeq1d 3436 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq2d 3437 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq12d 3438 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Theorem | ifbi 3439 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Theorem | ifbid 3440 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Theorem | ifbieq1d 3441 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
Theorem | ifbieq2i 3442 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq2d 3443 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3444 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3445 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3446 | Deduction version of nfif 3447. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3447 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifcldadc 3448 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
DECID | ||
Theorem | ifeq1dadc 3449 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | ifbothdadc 3450 | 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 3451 | 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 3452 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
DECID | ||
Theorem | eqifdc 3453 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
DECID | ||
Theorem | ifcldcd 3454 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifandc 3455 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
DECID | ||
Theorem | ifmdc 3456 | 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 3457 | 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 3458* | Soundness justification theorem for df-pw 3459. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-pw 3459* | 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 3460 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3461 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3462 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3463 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | selpw 3464* | Setvar variable membership in a power class (common case). See elpw 3463. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3465 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3466 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwb 3467 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Theorem | elpwid 3468 | An element of a power class is a subclass. Deduction form of elpwi 3466. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3469 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3470 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3471 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3472 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3473* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3474 | Extend class notation to include singleton. |
Syntax | cpr 3475 | Extend class notation to include unordered pair. |
Syntax | ctp 3476 | Extend class notation to include unordered triplet. |
Syntax | cop 3477 | Extend class notation to include ordered pair. |
Syntax | cotp 3478 | Extend class notation to include ordered triple. |
Theorem | snjust 3479* | Soundness justification theorem for df-sn 3480. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-sn 3480* | 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 3488. (Contributed by NM, 5-Aug-1993.) |
Definition | df-pr 3481 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3546. For a more traditional definition, but requiring a dummy variable, see dfpr2 3493. (Contributed by NM, 5-Aug-1993.) |
Definition | df-tp 3482 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-op 3483* |
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 3674 and opprc2 3675). For
Kuratowski's actual definition when the arguments are sets, see dfop 3651.
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3483 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3483 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 3484 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3485 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3486 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3487 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3488 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 3489 | 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 3490 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3491 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3492 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3493* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 3494 | 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 3495 | 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 3496 | 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 3497 | 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 3498 | 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 3499 | 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 3500 | 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.) |
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