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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb8mo 2301 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
 
Theoremcbveu 2302 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x ph  <->  E! y ps )
 
Theoremcbvmo 2303 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremmo3 2304* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo 2305* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y
 ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremeu2 2306* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu1 2307* 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.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) ) )
 
TheoremeuexALT 2308 Alternate proof of euex 2291. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E! x ph  ->  E. x ph )
 
Theoremeuor 2309 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
 |- 
 F/ x ph   =>    |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuorv 2310* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuor2 2311 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( -.  E. x ph 
 ->  ( E! x (
 ph  \/  ps )  <->  E! x ps ) )
 
Theoremsbmo 2312* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo4f 2313* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremmo4 2314* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremeu4 2315* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  ps )  ->  x  =  y ) ) )
 
Theoremmoim 2316 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
 
Theoremmoimi 2317 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x ps  ->  E* x ph )
 
TheoremmorimOLD 2318 TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2317 applied to ax-1 6, as demonstrated in the proof. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2319 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2320 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.)
 |-  ( ( E. x ph 
 /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )
 
Theoremmoan 2321 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
 |-  ( E* x ph  ->  E* x ( ps 
 /\  ph ) )
 
Theoremmoani 2322 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x ( ps 
 /\  ph )
 
Theoremmoor 2323 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
 |-  ( E* x (
 ph  \/  ps )  ->  E* x ph )
 
Theoremmooran1 2324 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 \/  E* x ps )  ->  E* x ( ph  /\ 
 ps ) )
 
Theoremmooran2 2325 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E* x (
 ph  \/  ps )  ->  ( E* x ph  /\ 
 E* x ps )
 )
 
Theoremmoanim 2326 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremeuan 2327 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmoanimv 2328* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoanmo 2329 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E* x ph )
 
Theoremmoaneu 2330 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
Theoremeuanv 2331* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmopick 2332 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupick 2333 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing  x such that 
ph is true, and there is also an  x (actually the same one) such that  ph and  ps are both true, then  ph implies  ps regardless of  x. 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.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupicka 2334 Version of eupick 2333 with closed formulas. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  A. x ( ph  ->  ps ) )
 
Theoremeupickb 2335 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( ( E! x ph 
 /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  <->  ps ) )
 
Theoremeupickbi 2336 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( E! x ph  ->  ( E. x (
 ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
 
Theoremmopick2 2337 "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 1725. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ph  /\  ps  /\  ch ) )
 
Theoremmoexex 2338 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theoremmoexexv 2339* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
 |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theorem2moex 2340 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E* x E. y ph  ->  A. y E* x ph )
 
Theorem2euex 2341 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x E. y ph  ->  E. y E! x ph )
 
Theorem2eumo 2342 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E* y ph  ->  E* x E! y ph )
 
Theorem2eu2ex 2343 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E! y ph  ->  E. x E. y ph )
 
Theorem2moswap 2344 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
 
Theorem2euswap 2345 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
 
Theorem2exeu 2346 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
 
Theorem2mo2 2347* This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair  x and  y". Note: this is not expressed by  E* x E* y ph). 2eu4 2355 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
 |-  ( ( E* x E. y ph  /\  E* y E. x ph )  <->  E. z E. w A. x A. y ( ph  ->  ( x  =  z 
 /\  y  =  w ) ) )
 
Theorem2mo 2348* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2moOLD 2349* Obsolete proof of 2mo 2348 as of 2-Nov-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 25-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2mos 2350* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w )
 ) 
 <-> 
 A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
 
Theorem2eu1 2351 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
 |-  ( A. x E* y ph  ->  ( E! x E! y ph  <->  ( E! x E. y ph  /\  E! y E. x ph )
 ) )
 
Theorem2eu1OLD 2352 Obsolete proof of 2eu1 2351 as of 11-Nov-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x E* y ph  ->  ( E! x E! y ph  <->  ( E! x E. y ph  /\  E! y E. x ph )
 ) )
 
Theorem2eu2 2353 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
 
Theorem2eu3 2354 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( A. x A. y ( E* x ph 
 \/  E* y ph )  ->  ( ( E! x E! y ph  /\  E! y E! x ph )  <->  ( E! x E. y ph  /\  E! y E. x ph ) ) )
 
Theorem2eu4 2355* This theorem provides us with a definition of double existential uniqueness ("exactly one 
x and exactly one  y"). Naively one might think (incorrectly) that it could be defined by  E! x E! y ph. See 2eu1 2351 for a condition under which the naive definition holds and 2exeu 2346 for a one-way implication. See 2eu5 2356 and 2eu8 2359 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu5 2356* An alternate definition of double existential uniqueness (see 2eu4 2355). A mistake sometimes made in the literature is to use  E! x E! y to mean "exactly one  x and exactly one  y." (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 
A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)
 |-  ( ( E! x E! y ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu6 2357* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E. z E. w A. x A. y ( ph  <->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2eu7 2358 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )
 
Theorem2eu8 2359 Two equivalent expressions for double existential uniqueness. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x E! y using 2eu7 2358. (Contributed by NM, 20-Feb-2005.)
 |-  ( E! x E! y ( E. x ph 
 /\  E. y ph )  <->  E! x E! y ( E! x ph  /\  E. y ph ) )
 
Theoremexists1 2360* 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 4613. (Contributed by NM, 5-Apr-2004.)
 |-  ( E! x  x  =  x  <->  A. x  x  =  y )
 
Theoremexists2 2361 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E. x ph 
 /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
 
1.7  Other axiomatizations related to classical predicate calculus
 
1.7.1  Aristotelian logic: Assertic syllogisms

Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems.

In antiquity Aristotelian logic and Stoic logic (see mptnan 1648) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order.

"There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation.

We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable  x. Our translation is essentially identical to the one use in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use  ph,  ps, and  ch in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26).

Expressions of the form "no  ph is  ps " are consistently translated as  A. x (
ph  ->  -.  ps ). These can also be expressed as  -.  E. x
( ph  /\  ps ), per alinexa 1709. We translate "all  ph is  ps " to  A. x (
ph  ->  ps ), "some  ph is  ps " to  E. x
( ph  /\  ps ), and "some  ph is not  ps " to  E. x
( ph  /\  -.  ps ). It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first.

In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as  x  =  A,  x  e.  A, or  x  C_  A. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like  x  e.  A instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion ( C_) and membership ( e.); this distinction was not directly made in Aristotle's work.

There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2366, celaront 2367, cesaro 2372, camestros 2373, felapton 2378, darapti 2379, calemos 2383, fesapo 2384, and bamalip 2385.

These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here.

Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus.

 
Theorembarbara 2362 "Barbara", one of the fundamental syllogisms of Aristotelian logic. All  ph is  ps, and all  ch is  ph, therefore all  ch is  ps. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as  A. x ( x  e.  H  ->  x  e.  M ) (all men are mortal) and  A. x ( x  =  S  ->  x  e.  H ) (Socrates is a man) therefore  A. x ( x  =  S  ->  x  e.  M ) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1748. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  ph )   =>    |-  A. x ( ch  ->  ps )
 
Theoremcelarent 2363 "Celarent", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ph, therefore no  ch is  ps. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ph )   =>    |-  A. x ( ch  ->  -.  ps )
 
Theoremdarii 2364 "Darii", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is  ph, therefore some  ch is  ps. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x ( ch  /\  ph )   =>    |-  E. x ( ch  /\  ps )
 
Theoremferio 2365 "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ph, therefore some  ch is not  ps. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ch  /\  ph )   =>    |-  E. x ( ch  /\  -.  ps )
 
Theorembarbari 2366 "Barbari", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is  ps. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremcelaront 2367 "Celaront", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremcesare 2368 "Cesare", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2363. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremcamestres 2369 "Camestres", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremfestino 2370 "Festino", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ps, therefore some  ch is not  ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ch  /\  ps )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theorembaroco 2371 "Baroco", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is not  ps, therefore some  ch is not  ph. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x ( ch  /\  -.  ps )   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcesaro 2372 "Cesaro", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcamestros 2373 "Camestros", one of the syllogisms of Aristotelian logic. All  ph is  ps, no  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremdatisi 2374 "Datisi", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x (
 ph  /\  ch )   =>    |-  E. x ( ch  /\  ps )
 
Theoremdisamis 2375 "Disamis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremferison 2376 "Ferison", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ph is  ch, therefore some  ch is not  ps. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ph  /\  ch )   =>    |-  E. x ( ch  /\  -.  ps )
 
Theorembocardo 2377 "Bocardo", one of the syllogisms of Aristotelian logic. Some  ph is not  ps, and all  ph is  ch, therefore some  ch is not  ps. (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2375; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
 |- 
 E. x ( ph  /\ 
 -.  ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremfelapton 2378 "Felapton", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  -. 
 ps )
 
Theoremdarapti 2379 "Darapti", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is  ps. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x (
 ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ps )
 
Theoremcalemes 2380 "Calemes", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ps is  ch, therefore no  ch is  ph. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremdimatis 2381 "Dimatis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ps is  ch, therefore some  ch is  ph. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2364 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x ( ps  ->  ch )   =>    |-  E. x ( ch  /\  ph )
 
Theoremfresison 2382 "Fresison", one of the syllogisms of Aristotelian logic. No  ph is  ps (PeM), and some  ps is  ch (MiS), therefore some  ch is not  ph (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ps  /\  ch )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theoremcalemos 2383 "Calemos", one of the syllogisms of Aristotelian logic. All  ph is  ps (PaM), no  ps is  ch (MeS), and  ch exist, therefore some  ch is not  ph (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremfesapo 2384 "Fesapo", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ps is  ch, and  ps exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ps   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theorembamalip 2385 "Bamalip", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ps is  ch, and  ph exist, therefore some  ch is  ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2366. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ph )
 
1.7.2  Intuitionistic logic

Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 417 or peirce 185. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 372 and df-ex 1661 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm.

The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1679, ax-11 1893, ax-gen 1666, ax-7 1840, ax-12 1906, ax-8 1871, ax-9 1873, and ax-5 1749.

In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.

 
Theoremaxia1 2386 Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( ( ph  /\  ps )  ->  ph )
 
Theoremaxia2 2387 Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( ( ph  /\  ps )  ->  ps )
 
Theoremaxia3 2388 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
 
Theoremaxin1 2389 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Theoremaxin2 2390 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theoremaxio 2391 Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremaxi4 2392 Specialization (intuitionistic logic axiom ax-4). This is just sp 1911 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
 |-  ( A. x ph  -> 
 ph )
 
Theoremaxi5r 2393 Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
 |-  ( ( A. x ph 
 ->  A. x ps )  ->  A. x ( A. x ph  ->  ps )
 )
 
Theoremaxial 2394  x is not free in  A. x ph (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremaxie1 2395  x is not free in  E. x ph (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremaxie2 2396 A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
 |-  ( A. x ( ps  ->  A. x ps )  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
 
Theoremaxi9 2397 Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1795 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
 |- 
 E. x  x  =  y
 
Theoremaxi10 2398 Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2105 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaxi12 2399 Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2102 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.)
 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremaxbnd 2400 Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2399 are fairly straightforward consequences of axc9 2102. But in intuitionistic logic, it is not easy to add the extra  A. x to axi12 2399 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.)
 |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y ) ) )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39884
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