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Theorem List for New Foundations Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhbeu1 1801 Bound-variable hypothesis builder for uniqueness.

Theoremhbeu 1802 Bound-variable hypothesis builder for "at most one." Note that and needn't be distinct (this makes the proof more difficult).

Theoremhbeud 1803 Deduction version of hbeu 1802.

Theoremsb8eu 1804 Variable substitution in uniqueness quantifier. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.)

Theoremcbveu 1805 Rule used to change bound variables, using implicit substitition. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 8-Jun-2011.)

Theoremeu1 1806* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.

Theoremmo 1807* Equivalent definitions of "there exists at most one."

Theoremeuex 1808 Existential uniqueness implies existence. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 1809* Existential uniqueness implies "at most one."

Theoremeu2 1810* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.

Theoremeu3 1811* An alternate way to express existential uniqueness.

Theoremeuor 1812 Introduce a disjunct into a uniqueness quantifier.

Theoremeuorv 1813* Introduce a disjunct into a uniqueness quantifier.

Theoremmo2 1814* Alternate definition of "at most one."

Theoremsbmo 1815* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo3 1816* 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.

Theoremmo4f 1817* "At most one" expressed using implicit substitution.

Theoremmo4 1818* "At most one" expressed using implicit substitution.

Theoremmobid 1819 Formula-building rule for "at most one" quantifier (deduction rule).

Theoremmobii 1820 Formula-building rule for "at most one" quantifier (inference rule).

Theoremhbmo1 1821 Bound-variable hypothesis builder for "at most one."

Theoremhbmo 1822 Bound-variable hypothesis builder for "at most one."

Theoremcbvmo 1823 Rule used to change bound variables, using implicit substitition. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 8-Jun-2011.)

Theoremeu5 1824 Uniqueness in terms of "at most one."

Theoremeu4 1825* Uniqueness using implicit substitution.

Theoremeumo 1826 Existential uniqueness implies "at most one."

Theoremeumoi 1827 "At most one" inferred from existential uniqueness.

Theoremexmoeu 1828 Existence in terms of "at most one" and uniqueness.

Theoremexmoeu2 1829 Existence implies "at most one" is equivalent to uniqueness.

Theoremmoabs 1830 Absorption of existence condition by "at most one."

Theoremexmo 1831 Something exists or at most one exists.

Theoremimmo 1832 "At most one" is preserved through implication (notice wff reversal).

Theoremimmoi 1833 "At most one" is preserved through implication (notice wff reversal).

Theoremmoimv 1834* Move antecedent outside of "at most one."

Theoremeuimmo 1835 Uniqueness implies "at most one" through implication.

Theoremeuim 1836 Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (The proof was shortened by Andrew Salmon, 14-Jun-2011.)

Theoremmoan 1837 "At most one" is still the case when a conjunct is added.

Theoremmoani 1838 "At most one" is still true when a conjunct is added.

Theoremmoor 1839 "At most one" is still the case when a disjunct is removed.

Theoremmooran1 1840 "At most one" imports disjunction to conjunction. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 1841 "At most one" exports disjunction to conjunction. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 1842 Introduction of a conjunct into "at most one" quantifier.

Theoremeuan 1843 Introduction of a conjunct into uniqueness quantifier. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanimv 1844* Introduction of a conjunct into "at most one" quantifier.

Theoremmoaneu 1845 Nested "at most one" and uniqueness quantifiers.

Theoremmoanmo 1846 Nested "at most one" quantifiers.

Theoremeuanv 1847* Introduction of a conjunct into uniqueness quantifier.

Theoremmopick 1848 "At most one" picks a variable value, eliminating an existential quantifier.

Theoremeupick 1849 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.

Theoremeupicka 1850 Version of eupick 1849 with closed formulas.

Theoremeupickb 1851 Existential uniqueness "pick" showing wff equivalence.

Theoremeupickbi 1852 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 1853 "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 1386. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuor2 1854 Introduce or eliminate a disjunct in a uniqueness quantifier. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexex 1855 "At most one" double quantification.

Theoremmoexexv 1856* "At most one" double quantification.

Theorem2moex 1857 Double quantification with "at most one."

Theorem2euex 1858 Double quantification with existential uniqueness. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 1859 Double quantification with existential uniqueness and "at most one."

Theorem2eu2ex 1860 Double existential uniqueness.

Theorem2moswap 1861 A condition allowing swap of "at most one" and existential quantifiers.

Theorem2euswap 1862 A condition allowing swap of uniqueness and existential quantifiers.

Theorem2exeu 1863 Double existential uniqueness implies double uniqueness quantification.

Theorem2mo 1864* Two equivalent expressions for double "at most one."

Theorem2mos 1865* Double "exists at most one", using implicit substitition.

Theorem2eu1 1866 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.

Theorem2eu2 1867 Double existential uniqueness.

Theorem2eu3 1868 Double existential uniqueness.

Theorem2eu4 1869* 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 1866 for a condition under which the naive definition holds and 2exeu 1863 for a one-way implication. See 2eu5 1870 and 2eu8 1873 for alternate definitions.

Theorem2eu5 1870* An alternate definition of double existential uniqueness (see 2eu4 1869). 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.")

Theorem2eu6 1871* Two equivalent expressions for double existential uniqueness.

Theorem2eu7 1872 Two equivalent expressions for double existential uniqueness.

Theorem2eu8 1873 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 1872.

Theoremeuequ1 1874* Equality has existential uniqueness. Special case of eueq1 2442 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)

Theoremexists1 1875* 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.

Theoremexists2 1876 A condition implying that at least two things exist. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

PART 2  NEW FOUNDATIONS (NF) SET THEORY

Here we introduce New Foundations set theory. We first introduce the axiom of extensionality in ax-ext 1877. We later add set construction axioms from {{Hailperin}}, such as ax-nin 3180, that are designed to implement the Stratification Axiom from {{Quine2}}.

We then introduce ordered pairs, relationships, and functions. Note that the definition of an ordered pair (in df-op 3674) is different than the Kuratowski ordered pair definition (in df-opk 2863) typically used in ZFC, because the Kuratowski definition is not type-level.

We conclude with orderings.

2.1.1  Introduce the Axiom of Extensionality

Axiomax-ext 1877* Axiom of Extensionality. An axiom of New Foundations set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461.

Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes , and equality is defined as . All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8.

To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1402 through ax-16 1606 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic.

General remarks: Our set theory axioms are presented using defined connectives (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , , , , and . It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so.

It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable in ax-ext 1877 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions (\$d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms. In practice, though, the theorems and proofs are essentially the same. The \$d restrictions make each of the the infinite axioms generated by the ax-ext 1877 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.

Theoremaxext2 1878* The Axiom of Extensionality (ax-ext 1877) restated so that it postulates the existence of a set given two arbitrary sets and . This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets.

Theoremaxext3 1879* A generalization of the Axiom of Extensionality in which and need not be distinct. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)

Theoremaxext4 1880* A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 1877 and df-cleq 1888.

Theorembm1.1 1881* Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462.

2.1.2  Class abstractions (a.k.a. class builders)

Syntaxcab 1882 Introduce the class builder or class abstraction notation ("the class of sets such that is true"). Our class variables , , etc. range over class builders (implicitly in the case of defined class terms such as df-nul 2727). Note that a set variable can be expressed as a class builder per theorem cvjust 1890, justifying the assignment of set variables to class variables via the use of cv 1397.

Definitiondf-clab 1883 Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. and need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, will have as a free variable, and " " is read "the class of all sets such that is true." We do not define in isolation but only as part of an expression that extends or "overloads" the relationship.

This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 1400, which extends or "overloads" the wel 1401 definition connecting set variables, requires that both sides of be a class. In df-cleq 1888 and df-clel 1891, we introduce a new kind of variable (class variable) that can substituted with expressions such as . In the present definition, the on the left-hand side is a set variable. Syntax definition cv 1397 allows us to substitute a set variable for a class variable: all sets are classes by cvjust 1890 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 1994 for a quick overview).

Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 2351.

Theoremabid 1884 Simplification of class abstraction notation when the free and bound variables are identical.

Theoremhbab1 1885* Bound-variable hypothesis builder for a class abstraction.

Theoremhbab 1886* Bound-variable hypothesis builder for a class abstraction.

Theoremhbabd 1887* Deduction form of bound-variable hypothesis builder hbab 1886.

Definitiondf-cleq 1888* Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce , which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 1880). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

We could avoid this complication by introducing a new symbol, say =2, in place of . This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition.. One of our theorems would then be =2 by invoking Extensionality.

However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality.

Theoremdfcleq 1889* The same as df-cleq 1888 with the hypothesis removed using the Axiom of Extensionality ax-ext 1877.

Theoremcvjust 1890* Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1397, which allows us to substitute a set variable for a class variable. See also cab 1882 and df-clab 1883. Note that this is not a rigorous justification, because cv 1397 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

Definitiondf-clel 1891* Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 1888 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 1888 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1732), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 1883. Alternate definitions of (but that require either or to be a set) are shown by clel2 2404, clel3 2406, and clel4 2407.

Theoremeqriv 1892* Infer equality of classes from equivalence of membership.

Theoremeqrdv 1893* Deduce equality of classes from equivalence of membership.

Theoremeqrdav 1894* Deduce equality of classes from an equivalence of membership that depends on the membership variable.

Theoremeqid 1895 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). (Thanks to Stefan Allan for this information.)

Theoremeqidd 1896 Class identity law with antecedent.

Theoremeqcom 1897 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.

Theoremeqcoms 1898 Inference applying commutative law for class equality to an antecedent.

Theoremeqcomi 1899 Inference from commutative law for class equality.

Theoremeqcomd 1900 Deduction from commutative law for class equality.

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