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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdifdifdir 3901 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)

Theoremuneqdifeq 3902 Two ways to say that and partition (when and don't overlap and is a part of ). (Contributed by FL, 17-Nov-2008.)

Theoremraldifeq 3903* Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)

Theoremr19.2z 3904* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1738). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)

Theoremr19.2zb 3905* A response to the notion that the condition can be removed in r19.2z 3904. Interestingly enough, does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)

Theoremr19.3rz 3906* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Theoremr19.28z 3907* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)

Theoremr19.3rzv 3908* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)

Theoremr19.9rzv 3909* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Theoremr19.28zv 3910* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)

Theoremr19.37zv 3911* Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)

Theoremr19.45zv 3912* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27z 3913* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)

Theoremr19.27zv 3914* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)

Theoremr19.36zv 3915* Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)

Theoremrzal 3916* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3917* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremralidm 3918* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3919 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenz 3920* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)

Theoremralf0 3921* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremraaan 3922* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3923* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3924* 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.)

Theoremsbcssg 3925 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)

2.1.15  "Weak deduction theorem" for set theory

In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps.

The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e. we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem.

We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs.

We define the conditional operator, if(P, A, B), as follows: if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P).

Lemma 1. A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality.

Lemma 2. A = if(P, A, B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality.

Here's a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme.

We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A.

The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However, it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem," i.e. deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example:

Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.)

Syntaxcif 3926 Extend class notation to include the conditional operator. See df-if 3927 for a description. (In older databases this was denoted "ded".)

Definitiondf-if 3927* Define the conditional operator. Read as "if then else ." See iftrue 3932 and iffalse 3935 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 older versions of this database, this operator was denoted "ded" and called the "deduction class.")

An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role, is a class variable in the hypothesis and is a class (usually a constant) that makes the hypothesis true when it is substituted for . See dedth 3978 for the main part of the weak deduction theorem, elimhyp 3985 to eliminate a hypothesis, and keephyp 3991 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.)

Theoremdfif2 3928* An alternate definition of the conditional operator df-if 3927 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)

Theoremdfif6 3929* An alternate definition of the conditional operator df-if 3927 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3930 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremifeq2 3931 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremiftrue 3932 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.)

Theoremiftruei 3933 Inference associated with iftrue 3932. (Contributed by BJ, 7-Oct-2018.)

Theoremiftrued 3934 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremiffalse 3935 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Theoremiffalsei 3936 Inference associated with iffalse 3935. (Contributed by BJ, 7-Oct-2018.)

Theoremiffalsed 3937 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremifnefalse 3938 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 3935 directly in this case. It happens, e.g., in oevn0 7167. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremifsb 3939 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)

Theoremdfif3 3940* Alternate definition of the conditional operator df-if 3927. 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 3941* Alternate definition of the conditional operator df-if 3927. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Theoremdfif5 3942* Alternate definition of the conditional operator df-if 3927. Note that is independent of i.e. a constant true or false (see also abvor0 3789). (Contributed by Gérard Lang, 18-Aug-2013.)

Theoremifeq12 3943 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)

Theoremifeq1d 3944 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq2d 3945 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq12d 3946 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Theoremifbi 3947 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Theoremifbid 3948 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)

Theoremifbieq1d 3949 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)

Theoremifbieq2i 3950 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq2d 3951 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq12i 3952 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Theoremifbieq12d 3953 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfifd 3954 Deduction version of nfif 3955. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremnfif 3955 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifeq1da 3956 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifeq2da 3957 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifclda 3958 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifeqda 3959 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

Theoremelimif 3960 Elimination of a conditional operator contained in a wff . (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)

Theoremifbothda 3961 A wff containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)

Theoremifboth 3962 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 3963 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Theoremeqif 3964 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremifval 3965 Another expression of the value of the predicate, analogous to eqif 3964. See also the more specialized iftrue 3932 and iffalse 3935. (Contributed by BJ, 6-Apr-2019.)

Theoremelif 3966 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremifel 3967 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)

Theoremifcl 3968 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)

Theoremifcld 3969 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)

Theoremifeqor 3970 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifnot 3971 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)

Theoremifan 3972 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremifor 3973 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorem2if2 3974 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)

Theoremifcomnan 3975 Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)

Theoremcsbif 3976 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)

TheoremcsbifgOLD 3977 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.) Obsolete as of 19-Aug-2018. Use csbif 3976 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdedth 3978 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 3985. If the inference has other hypotheses with class variable , these can be kept by assigning keephyp 3991 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpeuni/mmdeduction.html. (Contributed by NM, 15-May-1999.)

Theoremdedth2h 3979 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3982 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3978. (Contributed by NM, 15-May-1999.)

Theoremdedth3h 3980 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3979. (Contributed by NM, 15-May-1999.)

Theoremdedth4h 3981 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3979. (Contributed by NM, 16-May-1999.)

Theoremdedth2v 3982 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 3979 is simpler to use. See also comments in dedth 3978. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth3v 3983 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3982. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth4v 3984 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3982. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremelimhyp 3985 Eliminate a hypothesis containing class variable when it is known for a specific class . For more information, see comments in dedth 3978. (Contributed by NM, 15-May-1999.)

Theoremelimhyp2v 3986 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp3v 3987 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp4v 3988 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3978). (Contributed by NM, 16-Apr-2005.)

Theoremelimel 3989 Eliminate a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 15-May-1999.)

Theoremelimdhyp 3990 Version of elimhyp 3985 where the hypothesis is deduced from the final antecedent. See ghomgrplem 29007 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremkeephyp 3991 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 3992 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3978). (Contributed by NM, 16-Apr-2005.)

Theoremkeephyp3v 3993 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)

Theoremkeepel 3994 Keep a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 14-Aug-1999.)

Theoremifex 3995 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)

Theoremifexg 3996 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)

2.1.16  Power classes

Syntaxcpw 3997 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)

Theorempwjust 3998* Soundness justification theorem for df-pw 3999. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-pw 3999* 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 25128). We will later introduce the Axiom of Power Sets ax-pow 4615, which can be expressed in class notation per pwexg 4621. Still later we will prove, in hashpw 12476, 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, 24-Jun-1993.)

Theorempweq 4000 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37825
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