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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | disjpss 3601 | A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.) |
Theorem | undisj1 3602 | The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.) |
Theorem | undisj2 3603 | The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.) |
Theorem | ssindif0 3604 | Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.) |
Theorem | inelcm 3605 | The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.) |
Theorem | minel 3606 | A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) |
Theorem | undif4 3607 | Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | disjssun 3608 | Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ssdif0 3609 | Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) |
Theorem | vdif0 3610 | Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.) |
Theorem | pssdifn0 3611 | A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.) |
Theorem | pssdif 3612 | A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.) |
Theorem | ssnelpss 3613 | A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.) |
Theorem | ssnelpssd 3614 | Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3613. (Contributed by David Moews, 1-May-2017.) |
Theorem | pssnel 3615* | A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.) |
Theorem | difin0ss 3616 | Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Theorem | inssdif0 3617 | 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.) |
Theorem | difid 3618 | 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.) |
Theorem | difidALT 3619 | 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 3618. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | dif0 3620 | The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Theorem | 0dif 3621 | The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Theorem | disjdif 3622 | A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.) |
Theorem | difin0 3623 | 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.) |
Theorem | undifv 3624 | 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.) |
Theorem | undif1 3625 | Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.) |
Theorem | undif2 3626 | Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.) |
Theorem | undifabs 3627 | Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) |
Theorem | inundif 3628 | 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.) |
Theorem | difun2 3629 | Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.) |
Theorem | undif 3630 | Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.) |
Theorem | ssdifin0 3631 | 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.) |
Theorem | ssdifeq0 3632 | A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
Theorem | ssundif 3633 | A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.) |
Theorem | difcom 3634 | Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.) |
Theorem | pssdifcom1 3635 | Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) |
Theorem | pssdifcom2 3636 | Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) |
Theorem | difdifdir 3637 | Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.) |
Theorem | uneqdifeq 3638 | Two ways to say that and partition (when and don't overlap and is a part of ). (Contributed by FL, 17-Nov-2008.) |
Theorem | r19.2z 3639* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1659). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.) |
Theorem | r19.2zb 3640* | A response to the notion that the condition can be removed in r19.2z 3639. Interestingly enough, does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.) |
Theorem | r19.3rz 3641* | Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.) |
Theorem | r19.28z 3642* | 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.) |
Theorem | r19.3rzv 3643* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) |
Theorem | r19.9rzv 3644* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.) |
Theorem | r19.28zv 3645* | 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.) |
Theorem | r19.37zv 3646* | 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.) |
Theorem | r19.45zv 3647* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Theorem | r19.27z 3648* | 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.) |
Theorem | r19.27zv 3649* | 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.) |
Theorem | r19.36zv 3650* | 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.) |
Theorem | rzal 3651* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | rexn0 3652* | Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Theorem | ralidm 3653* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Theorem | ral0 3654 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
Theorem | rgenz 3655* | Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.) |
Theorem | ralf0 3656* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
Theorem | raaan 3657* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
Theorem | raaanv 3658* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Theorem | sbss 3659* | 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 | sbcss 3660 | 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 | sscon34 3661 | Contraposition law for subset. (Contributed by SF, 11-Mar-2015.) |
∼ ∼ | ||
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.) | ||
Syntax | cif 3662 | Extend class notation to include the conditional operator. See df-if 3663 for a description. (In older databases this was denoted "ded".) |
Definition | df-if 3663* |
Define the conditional operator. Read as "if
then
else ." See iftrue 3668 and iffalse 3669 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 3703 for the main part of the weak deduction theorem, elimhyp 3710 to eliminate a hypothesis, and keephyp 3716 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.) |
Theorem | dfif2 3664* | An alternate definition of the conditional operator df-if 3663 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.) |
Theorem | dfif6 3665* | An alternate definition of the conditional operator df-if 3663 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq1 3666 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | ifeq2 3667 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | iftrue 3668 | 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 | iffalse 3669 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
Theorem | ifnefalse 3670 | 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 3669 directly in this case. It happens, e.g., in oevn0 in set.mm. (Contributed by David A. Wheeler, 15-May-2015.) |
Theorem | ifsb 3671 | Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.) |
Theorem | dfif3 3672* | Alternate definition of the conditional operator df-if 3663. 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 | dfif4 3673* | Alternate definition of the conditional operator df-if 3663. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) |
Theorem | dfif5 3674* | Alternate definition of the conditional operator df-if 3663. Note that is independent of i.e. a constant true or false (see also abvor0 3567). (Contributed by Gérard Lang, 18-Aug-2013.) |
Theorem | ifeq12 3675 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
Theorem | ifeq1d 3676 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq2d 3677 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Theorem | ifeq12d 3678 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
Theorem | ifbi 3679 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Theorem | ifbid 3680 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
Theorem | ifbieq2i 3681 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq2d 3682 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3683 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3684 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3685 | Deduction version of nfif 3686. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3686 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifeq1da 3687 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | ifeq2da 3688 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | ifclda 3689 | Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | csbifg 3690 | Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Theorem | elimif 3691 | Elimination of a conditional operator contained in a wff . (Contributed by NM, 15-Feb-2005.) |
Theorem | ifbothda 3692 | A wff containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.) |
Theorem | ifboth 3693 | 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.) |
Theorem | ifid 3694 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
Theorem | eqif 3695 | Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.) |
Theorem | elif 3696 | Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
Theorem | ifel 3697 | Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
Theorem | ifcl 3698 | Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.) |
Theorem | ifeqor 3699 | The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifnot 3700 | Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.) |
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