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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | mercolem5 1501 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | mercolem6 1502 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | mercolem7 1503 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | mercolem8 1504 | |

Theorem | re1tbw1 1505 | tbw-ax1 1460 rederived from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re1tbw2 1506 | tbw-ax2 1461 rederived from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re1tbw3 1507 | tbw-ax3 1462 rederived from merco2 1496. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re1tbw4 1508 |
tbw-ax4 1463 rederived from merco2 1496.
This theorem, along with re1tbw1 1505, re1tbw2 1506, and re1tbw3 1507, shows that merco2 1496, along with ax-mp 10, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

1.4.9 Derive the Lukasiewicz axioms from the The
Russell-Bernays Axioms | ||

Theorem | rb-bijust 1509 | Justification for rb-imdf 1510. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rb-imdf 1510 | The definition of implication, in terms of and . (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | anmp 1511 | Modus ponens for axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rb-ax1 1512 | The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rb-ax2 1513 | The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rb-ax3 1514 | The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rb-ax4 1515 | The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rbsyl 1516 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rblem1 1517 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rblem2 1518 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rblem3 1519 | |

Theorem | rblem4 1520 | |

Theorem | rblem5 1521 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rblem6 1522 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | rblem7 1523 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re1axmp 1524 | ax-mp 10 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re2luk1 1525 | luk-1 1415 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re2luk2 1526 | luk-2 1416 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | re2luk3 1527 |
luk-3 1417 derived from Russell-Bernays'.
This theorem, along with re1axmp 1524, re2luk1 1525, and re2luk2 1526 shows that rb-ax1 1512, rb-ax2 1513, rb-ax3 1514, and rb-ax4 1515, along with anmp 1511, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |

1.4.10 Stoic logic indemonstrables (Chrysippus
of Soli)
The Greek Stoics developed a system of logic.
The Stoic Chrysippus, in particular, was often considered one of the greatest
logicians of antiquity.
Stoic logic is different from Aristotle's system, since it focuses
on propositional logic,
though later thinkers did combine the systems of the Stoics with Aristotle.
Jan Lukasiewicz reports,
"For anybody familiar with mathematical logic it is self-evident
that the Stoic dialectic is the ancient form of modern propositional
logic"
( A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 10, modus tollendo tollens (modus tollens) mto 169, modus ponendo tollens I mpto1 1528, modus ponendo tollens II mpto2 1529, and modus tollendo ponens (exclusive-or version) mtp-xor 1530. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtp-xor 1530 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtp-or 1531. This set of indemonstrables is not the entire system of Stoic logic. | ||

Theorem | mpto1 1528 | Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mpto2 1529) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) |

Theorem | mpto2 1529 | Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) |

Theorem | mtp-xor 1530 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, one of the five "indemonstrables" in Stoic logic. The rule says, "if is not true, and either or (exclusively) are true, then must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtp-or 1531. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1529, that is, it is exclusive-or df-xor 1301), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mpto2 1529), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) |

Theorem | mtp-or 1531 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1530, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if is not true, and or (or both) are true, then must be true." An alternative phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) |

1.5 Predicate calculus with equality: Tarski's
system S2 (1 rule, 6 schemes)Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "set variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all") in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1534) which is defined in terms of universal quantification.
Our axioms are really axiom Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our ax-4 2078) since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1664 below). Theorem spw 1664 can be used to prove any instance of ax-4 2078 having no wff metavariables and mutually distinct set variables. However, it seems that ax-4 2078 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. ax-4 2078 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove ax-4 2078 as theorem ax4 1720 using the auxiliary axioms that make our system metalogically complete. Our version of Tarski's system S2 consists of propositional calculus plus ax-gen 1538, ax-5 1549, ax-17 1608, ax-9 1641, ax-8 1648, ax-13 1690, and ax-14 1692. The last 3 are equality axioms that represent 3 sub-schemes of Tarski's scheme B8. Due to its side-condition ("where is an atomic formula and is obtained by replacing an occurrence of the variable by the variable "), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-8 1648, ax-13 1690, and ax-14 1692 are sufficient for set theory. Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of free variable and proper substitution) is arguably easier for a non-logician human to follow step by step in a proof. However, in our system that derives schemes (rather than object language theorems) from other schemes, Tarski's S2 is not complete. For example, we cannot derive scheme ax-4 2078, even though (using spw 1664) we can derive all instances of it that don't involve wff metavariables or bundled set metavariables. (Two set metavariables are "bundled" if they can be substituted with the same set metavariable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-6 1707, ax-7 1712, ax-12 1869, and ax-11 1719 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "metalogical completeness," allowing us to prove directly (instead of, say, induction on formula length) all possible schemes that can be expressed in our language. | ||

1.5.1 Universal quantifier; define "exists" and
"not free" | ||

Syntax | wal 1532 | Extend wff definition to include the universal quantifier ('for all'). is read " (phi) is true for all ." Typically, in its final application would be replaced with a wff containing a (free) occurrence of the variable , for example . In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of . When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |

Syntax | wex 1533 | Extend wff definition to include the existential quantifier ("there exists"). |

Definition | df-ex 1534 | Define existential quantification. means "there exists at least one set such that is true." Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) |

Theorem | alnex 1535 | Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |

Syntax | wnf 1536 | Extend wff definition to include the not-free predicate. |

Definition | df-nf 1537 |
Define the not-free predicate for wffs. This is read " is not free
in ".
Not-free means that the value of cannot affect the
value of ,
e.g., any occurrence of in
is effectively
bound by a "for all" or something that expands to one (such as
"there
exists"). In particular, substitution for a variable not free in a
wff
does not affect its value (sbf 1936). An example of where this is used is
stdpc5 1797. See nf2 1802 for an alternative definition which
does not involve
nested quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, is effectively not free in the bare expression (see nfequid 2102), even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result). This predicate only applies to wffs. See df-nfc 2409 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.) |

1.5.2 Rule scheme ax-gen
(Generalization) | ||

Axiom | ax-gen 1538 | Rule of Generalization. The postulated inference rule of pure predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved , we can conclude or even . Theorem allt 24247 shows the special case . Theorem spi 1742 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.) |

Theorem | gen2 1539 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |

Theorem | mpg 1540 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |

Theorem | mpgbi 1541 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |

Theorem | mpgbir 1542 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |

Theorem | nfi 1543 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | hbth 1544 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form from smaller formulas of this form. These are useful for constructing hypotheses that state " is (effectively) not free in ." (Contributed by NM, 5-Aug-1993.) |

Theorem | nfth 1545 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nftru 1546 | The true constant has no free variables. (This can also be proven in one step with nfv 1610, but this proof does not use ax-17 1608.) (Contributed by Mario Carneiro, 6-Oct-2016.) |

Theorem | nex 1547 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |

Theorem | nfnth 1548 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |

1.5.3 Axiom scheme ax-5 (Quantified
Implication) | ||

Axiom | ax-5 1549 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |

Theorem | alim 1550 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |

Theorem | alimi 1551 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |

Theorem | 2alimi 1552 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |

Theorem | al2imi 1553 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |

Theorem | alanimi 1554 | Variant of al2imi 1553 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |

Theorem | alimdh 1555 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |

Theorem | albi 1556 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | alrimih 1557 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | albii 1558 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |

Theorem | 2albii 1559 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |

Theorem | hbxfrbi 1560 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2387 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |

Theorem | nfbii 1561 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfxfr 1562 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | nfxfrd 1563 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |

Theorem | alex 1564 | Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |

Theorem | 2nalexn 1565 | Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |

Theorem | exnal 1566 | Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | exim 1567 | Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |

Theorem | eximi 1568 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |

Theorem | 2eximi 1569 | Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |

Theorem | alinexa 1570 | A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |

Theorem | alexn 1571 | A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.) |

Theorem | 2exnexn 1572 | Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.) |

Theorem | exbi 1573 | Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | exbii 1574 | Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.) |

Theorem | 2exbii 1575 | Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |

Theorem | 3exbii 1576 | Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.) |

Theorem | exanali 1577 | A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.) |

Theorem | exancom 1578 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |

Theorem | alrimdh 1579 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |

Theorem | eximdh 1580 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.) |

Theorem | nexdh 1581 | Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |

Theorem | albidh 1582 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |

Theorem | exbidh 1583 | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |

Theorem | exsimpl 1584 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | 19.26 1585 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |

Theorem | 19.26-2 1586 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |

Theorem | 19.26-3an 1587 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |

Theorem | 19.29 1588 | Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |

Theorem | 19.29r 1589 | Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |

Theorem | 19.29r2 1590 | Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |

Theorem | 19.29x 1591 | Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |

Theorem | 19.35 1592 | Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |

Theorem | 19.35i 1593 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.35ri 1594 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.25 1595 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.30 1596 | Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |

Theorem | 19.43 1597 | Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |

Theorem | 19.43OLD 1598 | Obsolete proof of 19.43 1597 as of 3-May-2016. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | 19.33 1599 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |

Theorem | 19.33b 1600 | The antecedent provides a condition implying the converse of 19.33 1599. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.) |

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