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Theorem List for New Foundations Explorer - 2201-2300   *Has distinct variable group(s)
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Statement

Theoremax11v2-o 2201* Recovery of ax-11o 2141 from ax11v 2096 without using ax-11o 2141. The hypothesis is even weaker than ax11v 2096, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2141. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11a2-o 2202* Derive ax-11o 2141 from a hypothesis in the form of ax-11 1746, without using ax-11 1746 or ax-11o 2141. The hypothesis is even weaker than ax-11 1746, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1746, if we also hvae ax-10o 2139 which this proof uses . As Theorem ax11 2155 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2140 instead of ax-10o 2139. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10o-o 2203 Show that ax-10o 2139 can be derived from ax-10 2140. An open problem is whether this theorem can be derived from ax-10 2140 and the others when ax-11 1746 is replaced with ax-11o 2141. See Theorem ax10from10o 2177 for the rederivation of ax-10 2140 from ax10o 1952.

Normally, ax10o 1952 should be used rather than ax-10o 2139 or ax10o-o 2203, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

1.7  Existential uniqueness

Syntaxweu 2204 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2205 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2206* A soundness justification theorem for df-eu 2208, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2207 for a proof that provides an example of how it can be achieved through the use of dvelim 2016. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

TheoremeujustALT 2207* A soundness justification theorem for df-eu 2208, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. While this isn't strictly necessary for soundness, the proof provides an example of how it can be achieved through the use of dvelim 2016. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Definitiondf-eu 2208* Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2225, eu2 2229, eu3 2230, and eu5 2242 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 2287). (Contributed by NM, 12-Aug-1993.)

Definitiondf-mo 2209 Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2235. For other possible definitions see mo2 2233 and mo4 2237. (Contributed by NM, 8-Mar-1995.)

Theoremeuf 2210* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubid 2211 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 2212* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 2213 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfeu1 2214 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2215 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfeud2 2216 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfmod2 2217 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeud 2218 Deduction version of nfeu 2220. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmod 2219 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeu 2220 Bound-variable hypothesis builder for "at most one." Note that and needn't be distinct (this makes the proof more difficult). (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo 2221 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremsb8eu 2222 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 2223 Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2224 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 2225* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremmo 2226* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeuex 2227 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 2228* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeu2 2229* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3 2230* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeuor 2231 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 2232* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2 2233* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremsbmo 2234* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo3 2235* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)

Theoremmo4f 2236* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 2237* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremmobid 2238 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 2239* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 2240 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremcbvmo 2241 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremeu5 2242 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeu4 2243* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeumo 2244 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeumoi 2245 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremexmoeu 2246 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremexmoeu2 2247 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 2248 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmo 2249 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)

Theoremmoim 2250 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 2251 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmorimv 2252* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Theoremeuimmo 2253 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 2254 Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremmoan 2255 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 2256 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 2257 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 2258 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 2259 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 2260 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Theoremeuan 2261 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanimv 2262* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoaneu 2263 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoanmo 2264 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremeuanv 2265* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmopick 2266 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Theoremeupick 2267 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)

Theoremeupicka 2268 Version of eupick 2267 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 2269 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Theoremeupickbi 2270 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 2271 "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1609. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuor2 2272 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexex 2273 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.)

Theoremmoexexv 2274* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)

Theorem2moex 2275 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2euex 2276 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 2277 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2eu2ex 2278 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2moswap 2279 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2euswap 2280 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2exeu 2281 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)

Theorem2mo 2282* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorem2mos 2283* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)

Theorem2eu1 2284 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)

Theorem2eu2 2285 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2eu3 2286 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2eu4 2287* This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 2284 for a condition under which the naive definition holds and 2exeu 2281 for a one-way implication. See 2eu5 2288 and 2eu8 2291 for alternate definitions. (Contributed by NM, 3-Dec-2001.)

Theorem2eu5 2288* An alternate definition of double existential uniqueness (see 2eu4 2287). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.") (Contributed by NM, 26-Oct-2003.)

Theorem2eu6 2289* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorem2eu7 2290 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Theorem2eu8 2291 Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 2290. (Contributed by NM, 20-Feb-2005.)

Theoremeuequ1 2292* Equality has existential uniqueness. Special case of eueq1 3009 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)

Theoremexists1 2293* Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru in set.mm. (Contributed by NM, 5-Apr-2004.)

Theoremexists2 2294 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

1.8  Other axiomatizations related to classical predicate calculus

1.8.1  Predicate calculus with all distinct variables

Axiomax-7d 2295* Distinct variable version of ax-7 1734. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-8d 2296* Distinct variable version of ax-8 1675. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-9d1 2297 Distinct variable version of ax9 1949, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-9d2 2298* Distinct variable version of ax9 1949, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-10d 2299* Distinct variable version of ax10 1944. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-11d 2300* Distinct variable version of ax-11 1746. (Contributed by Mario Carneiro, 14-Aug-2015.)

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