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

Theorem0dif 3701 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theoremdisjdif 3702 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3703 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremundifv 3704 The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)

Theoremundif1 3705 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3702). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundif2 3706 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3702). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)

Theoremundifabs 3707 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Theoreminundif 3708 The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifun2 3709 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundif 3710 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)

Theoremssdifin0 3711 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremssdifeq0 3712 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Theoremssundif 3713 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)

Theoremdifcom 3714 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)

Theorempssdifcom1 3715 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)

Theorempssdifcom2 3716 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)

Theoremdifdifdir 3717 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)

Theoremuneqdifeq 3718 Two ways to say that and partition (when and don't overlap and is a part of ). (Contributed by FL, 17-Nov-2008.)

Theoremr19.2z 3719* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1672). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)

Theoremr19.2zb 3720* A response to the notion that the condition can be removed in r19.2z 3719. Interestingly enough, does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)

Theoremr19.3rz 3721* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Theoremr19.28z 3722* 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 3723* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)

Theoremr19.9rzv 3724* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Theoremr19.28zv 3725* 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 3726* 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 3727* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27z 3728* 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 3729* 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 3730* 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 3731* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3732* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremralidm 3733* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3734 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenz 3735* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)

Theoremralf0 3736* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremraaan 3737* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3738* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3739* 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.)

Theoremsbcss 3740 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 3741 Extend class notation to include the conditional operator. See df-if 3742 for a description. (In older databases this was denoted "ded".)

Definitiondf-if 3742* Define the conditional operator. Read as "if then else ." See iftrue 3747 and iffalse 3748 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 3782 for the main part of the weak deduction theorem, elimhyp 3789 to eliminate a hypothesis, and keephyp 3795 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 3743* An alternate definition of the conditional operator df-if 3742 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)

Theoremdfif6 3744* An alternate definition of the conditional operator df-if 3742 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3745 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremifeq2 3746 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremiftrue 3747 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 3748 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Theoremifnefalse 3749 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 3748 directly in this case. It happens, e.g., in oevn0 6761. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremifsb 3750 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)

Theoremdfif3 3751* Alternate definition of the conditional operator df-if 3742. 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 3752* Alternate definition of the conditional operator df-if 3742. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Theoremdfif5 3753* Alternate definition of the conditional operator df-if 3742. Note that is independent of i.e. a constant true or false (see also abvor0 3647). (Contributed by Gérard Lang, 18-Aug-2013.)

Theoremifeq12 3754 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)

Theoremifeq1d 3755 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq2d 3756 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq12d 3757 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Theoremifbi 3758 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Theoremifbid 3759 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)

Theoremifbieq2i 3760 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq2d 3761 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq12i 3762 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Theoremifbieq12d 3763 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfifd 3764 Deduction version of nfif 3765. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremnfif 3765 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifeq1da 3766 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifeq2da 3767 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremifclda 3768 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremcsbifg 3769 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelimif 3770 Elimination of a conditional operator contained in a wff . (Contributed by NM, 15-Feb-2005.)

Theoremifbothda 3771 A wff containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)

Theoremifboth 3772 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 3773 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Theoremeqif 3774 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremelif 3775 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Theoremifel 3776 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)

Theoremifcl 3777 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)

Theoremifeqor 3778 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifnot 3779 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)

Theoremifan 3780 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremifor 3781 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremdedth 3782 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 3789. If the inference has other hypotheses with class variable , these can be kept by assigning keephyp 3795 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpeuni/mmdeduction.html. (Contributed by NM, 15-May-1999.)

Theoremdedth2h 3783 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3786 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3782. (Contributed by NM, 15-May-1999.)

Theoremdedth3h 3784 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3783. (Contributed by NM, 15-May-1999.)

Theoremdedth4h 3785 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3783. (Contributed by NM, 16-May-1999.)

Theoremdedth2v 3786 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 3783 is simpler to use. See also comments in dedth 3782. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth3v 3787 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3786. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremdedth4v 3788 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3786. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Theoremelimhyp 3789 Eliminate a hypothesis containing class variable when it is known for a specific class . For more information, see comments in dedth 3782. (Contributed by NM, 15-May-1999.)

Theoremelimhyp2v 3790 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp3v 3791 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

Theoremelimhyp4v 3792 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3782). (Contributed by NM, 16-Apr-2005.)

Theoremelimel 3793 Eliminate a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 15-May-1999.)

Theoremelimdhyp 3794 Version of elimhyp 3789 where the hypothesis is deduced from the final antecedent. See ghomgrplem 25102 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremkeephyp 3795 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 3796 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3782). (Contributed by NM, 16-Apr-2005.)

Theoremkeephyp3v 3797 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)

Theoremkeepel 3798 Keep a membership hypothesis for weak deduction theorem, when special case is provable. (Contributed by NM, 14-Aug-1999.)

Theoremifex 3799 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)

Theoremifexg 3800 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)

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