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

Theoremneq0r 3301* An inhabited class is nonempty. See n0rf 3299 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremreximdva0m 3302* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0mmoeu 3303* A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremrex0 3304 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3305* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3306* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theoremnotm0 3307* A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)

Theoremnel0 3308* From the general negation of membership in , infer that is the empty set. (Contributed by BJ, 6-Oct-2018.)

Theorem0el 3309* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0dc 3310* The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
DECID

Theoremabn0r 3311 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremabn0m 3312* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)

Theoremrabn0r 3313 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0m 3314* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Theoremrab0 3315 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 3316 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremabeq0 3317 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theoremrabxmdc 3318* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID

Theoremrabnc 3319* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3320 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3321 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.)

Theorem0in 3322 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoreminv1 3323 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 3324 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 3325 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3326 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3327 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3328 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3329 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3330 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3331* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremcsbprc 3332 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)

Theoremun00 3333 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3334 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.)

Theoremdisj 3335* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3336* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3337* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3338 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 3339 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3340 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3341 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3342 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremssdisj 3343 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremundisj1 3344 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3345 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0im 3346 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3347 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3348 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3349 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3350 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0im 3351 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 3352 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifrab0eqim 3353* 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.)

Theoreminssdif0im 3354 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifid 3355 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 3356 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 3355. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdif0 3357 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 3358 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 3359 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3360 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 3361 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 3362 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 3363 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Theoreminundifss 3364 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.)

Theoremdisjdif2 3365 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)

Theoremdifun2 3366 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundifss 3367 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremssdifin0 3368 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 3369 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Theoremssundifim 3370 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 3371 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 3372 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 3373* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1575). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.3rm 3374* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremr19.28m 3375* 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 3376* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremr19.9rmv 3377* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.28mv 3378* 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 3379* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.44mv 3380* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27m 3381* 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 3382* 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 3383* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3384* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3385). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremrexm 3385* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Theoremralidm 3386* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3387 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenm 3388* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremralf0 3389* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremralm 3390 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaanlem 3391* Special case of raaan 3392 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaan 3392* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3393* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3394* 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 3395 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)

Theoremdcun 3396 The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
DECID        DECID        DECID

2.1.15  Conditional operator

Syntaxcif 3397 Extend class notation to include the conditional operator. See df-if 3398 for a description. (In older databases this was denoted "ded".)

Definitiondf-if 3398* Define the conditional operator. Read as "if then else ." See iftrue 3402 and iffalse 3405 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 782). (Contributed by NM, 15-May-1999.)

Theoremdfif6 3399* An alternate definition of the conditional operator df-if 3398 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremifeq1 3400 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

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