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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdisjne 3301 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3302 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3303 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremdisj4im 3304 A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)

Theoremssdisj 3305 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremdisjpss 3306 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

Theoremundisj1 3307 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3308 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0im 3309 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3310 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3311 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3312 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3313 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0im 3314 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)

Theoremvdif0im 3315 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifrab0eqim 3316* 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 Jim Kingdon, 3-Aug-2018.)

Theoremssnelpss 3317 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)

Theoremssnelpssd 3318 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3317. (Contributed by David Moews, 1-May-2017.)

Theoreminssdif0im 3319 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifid 3320 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 3321 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 3320. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdif0 3322 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 3323 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 3324 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3325 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.)

Theoremundif1ss 3326 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundif2ss 3327 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundifabs 3328 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Theoreminundifss 3329 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremdifun2 3330 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundifss 3331 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremssdifin0 3332 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 3333 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Theoremssundifim 3334 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremdifdifdirss 3335 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremuneqdifeqim 3336 Two ways that and can "partition" (when and don't overlap and is a part of ). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremr19.2m 3337* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1545). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.3rm 3338* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremr19.28m 3339* 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, 5-Aug-2018.)

Theoremr19.3rmv 3340* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremr19.9rmv 3341* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.28mv 3342* 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.)

Theoremr19.45mv 3343* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27m 3344* 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.)

Theoremr19.27mv 3345* 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.)

Theoremrzal 3346* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3347* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3348). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremrexm 3348* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Theoremralidm 3349* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3350 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenm 3351* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremralf0 3352* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremralm 3353 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaanlem 3354* Special case of raaan 3355 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaan 3355* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3356* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3357* 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 3358 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  Conditional operator

Syntaxcif 3359 Extend class notation to include the conditional operator. See df-if 3360 for a description. (In older databases this was denoted "ded".)

Definitiondf-if 3360* Define the conditional operator. Read as "if then else ." See iftrue 3364 and iffalse 3367 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 754). (Contributed by NM, 15-May-1999.)

Theoremdfif6 3361* An alternate definition of the conditional operator df-if 3360 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3362 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremifeq2 3363 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremiftrue 3364 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 3365 Inference associated with iftrue 3364. (Contributed by BJ, 7-Oct-2018.)

Theoremiftrued 3366 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremiffalse 3367 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Theoremiffalsei 3368 Inference associated with iffalse 3367. (Contributed by BJ, 7-Oct-2018.)

Theoremiffalsed 3369 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremifnefalse 3370 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 3367 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremdfif3 3371* Alternate definition of the conditional operator df-if 3360. 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.)

Theoremifeq12 3372 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)

Theoremifeq1d 3373 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq2d 3374 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Theoremifeq12d 3375 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Theoremifbi 3376 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Theoremifbid 3377 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)

Theoremifbieq1d 3378 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)

Theoremifbieq2i 3379 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq2d 3380 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq12i 3381 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Theoremifbieq12d 3382 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfifd 3383 Deduction version of nfif 3384. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremnfif 3384 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifbothdc 3385 A wff containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
DECID

Theoremifcldcd 3386 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
DECID

2.1.16  Power classes

Syntaxcpw 3387 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)

Theorempwjust 3388* Soundness justification theorem for df-pw 3389. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-pw 3389* 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.)

Theorempweq 3390 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Theorempweqi 3391 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)

Theorempweqd 3392 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)

Theoremelpw 3393 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Theoremselpw 3394* Setvar variable membership in a power class (common case). See elpw 3393. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelpwg 3395 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)

Theoremelpwi 3396 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Theoremelpwid 3397 An element of a power class is a subclass. Deduction form of elpwi 3396. (Contributed by David Moews, 1-May-2017.)

Theoremelelpwi 3398 If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.)

Theoremnfpw 3399 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theorempwidg 3400 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

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