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Theorem List for Metamath Proof Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0el 3601* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0 3602* The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)

Theoremabn0 3603 Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremrabn0 3604 Non-empty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)

Theoremrab0 3605 Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrabeq0 3606 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremrabxm 3607* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremrabnc 3608* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3609 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3610 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminv1 3611 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theoremunv 3612 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theorem0ss 3613 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3614 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3615 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3616 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3617 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3618 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3619* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremun00 3620 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3621 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorem0pss 3622 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)

Theoremnpss0 3623 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempssv 3624 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)

Theoremdisj 3625* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3626* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3627* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3628 Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj3 3629 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3630 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3631 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3632 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremdisj4 3633 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)

Theoremssdisj 3634 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremdisjpss 3635 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

Theoremundisj1 3636 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3637 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0 3638 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3639 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3640 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3641 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3642 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0 3643 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Theoremvdif0 3644 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)

Theoremdifrab0eq 3645* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Theorempssdifn0 3646 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)

Theorempssdif 3647 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)

Theoremssnelpss 3648 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)

Theoremssnelpssd 3649 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3648. (Contributed by David Moews, 1-May-2017.)

Theorempssnel 3650* A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)

Theoremdifin0ss 3651 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Theoreminssdif0 3652 Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Theoremdifid 3653 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)

TheoremdifidALT 3654 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3653. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdif0 3655 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theorem0dif 3656 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 3657 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3658 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 3659 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 3660 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3657). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundif2 3661 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3657). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)

Theoremundifabs 3662 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Theoreminundif 3663 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 3664 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundif 3665 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)

Theoremssdifin0 3666 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 3667 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Theoremssundif 3668 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)

Theoremdifcom 3669 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)

Theorempssdifcom1 3670 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)

Theorempssdifcom2 3671 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)

Theoremdifdifdir 3672 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)

Theoremuneqdifeq 3673 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 3674* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1667). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)

Theoremr19.2zb 3675* A response to the notion that the condition can be removed in r19.2z 3674. Interestingly enough, does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)

Theoremr19.3rz 3676* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Theoremr19.28z 3677* 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 3678* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)

Theoremr19.9rzv 3679* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Theoremr19.28zv 3680* 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 3681* 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 3682* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27z 3683* 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 3684* 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 3685* 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 3686* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3687* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremralidm 3688* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3689 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenz 3690* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)

Theoremralf0 3691* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremraaan 3692* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3693* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3694* 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 3695 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 3696 Extend class notation to include the conditional operator. See df-if 3697 for a description. (In older databases this was denoted "ded".)

Definitiondf-if 3697* Define the conditional operator. Read as "if then else ." See iftrue 3702 and iffalse 3703 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 3737 for the main part of the weak deduction theorem, elimhyp 3744 to eliminate a hypothesis, and keephyp 3750 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 3698* An alternate definition of the conditional operator df-if 3697 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)

Theoremdfif6 3699* An alternate definition of the conditional operator df-if 3697 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3700 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

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