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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstdpc6 1801 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1802.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
 |- 
 A. x  x  =  x
 
Theoremstdpc7 1802 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1801.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph (
x ,  x )  ->  ph ( x ,  y ) ), provided that  y is free for  x in  ph ( x ,  x
)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  ->  ph )
 )
 
Theoremequtr2 1803 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
 
Theoremequviniv 1804* A specialized version of equvini 2088 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  y  =  z ) )
 
Theoremequvin 1805* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1838, ax-13 2000. (Revised by Wolf Lammen, 10-Jun-2019.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremax13b 1806 Two equivalent ways of expressing ax-13 2000. See the comment for ax-13 2000. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.)
 |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) ) )
 
Theoremspfw 1807* Weak version of sp 1860. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremspw 1808* Weak version of the specialization scheme sp 1860. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1860 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1860 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1832 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1860 are spfw 1807 (minimal distinct variable requirements), spnfw 1786 (when  x is not free in  -.  ph), spvw 1757 (when  x does not appear in  ph), sptruw 1631 (when  ph is true), and spfalw 1787 (when  ph is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremcbvalw 1809* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvalvw 1810* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexvw 1811* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremalcomiw 1812* Weak version of alcom 1846. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremhbn1fw 1813* Weak version of ax-10 1838 from which we can prove any ax-10 1838 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps 
 ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( -.  A. y ps  ->  A. x  -.  A. y ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1w 1814* Weak version of hbn1 1839. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremhba1w 1815* Weak version of hba1 1897. See comments for ax10w 1826. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremhbe1w 1816* Weak version of hbe1 1840. See comments for ax10w 1826. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph 
 ->  A. x E. x ph )
 
Theoremhbalw 1817* Weak version of hbal 1845. Uses only Tarski's FOL axiom schemes. Unlike hbal 1845, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremcbvaev 1818* Change bound variable in an equality with a dv condition. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
1.4.9  Membership predicate
 
Syntaxwcel 1819 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  e.  B here is to allow us to express i.e. "prove" the wel 1820 of predicate calculus in terms of the wcel 1819 of set theory, so that we don't "overload" the  e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2443 for more information on the set theory usage of wcel 1819.)

 wff  A  e.  B
 
Theoremwel 1820 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " x is an element of  y," " x is a member of  y," " x belongs to  y," or " y contains  x." Note: The phrase " y includes  x " means " x is a subset of  y;" to use it also for  x  e.  y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol  e. (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for  e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1820 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1819. This lets us avoid overloading the  e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1820 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1819. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

 wff  x  e.  y
 
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)
 
Axiomax-8 1821 Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate 
e., which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 30-Jun-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  ->  y  e.  z ) )
 
Theoremelequ1 1822 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
 |-  ( x  =  y 
 ->  ( x  e.  z  <->  y  e.  z ) )
 
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)
 
Axiomax-9 1823 Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate 
e., which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 21-Jun-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  ->  z  e.  y ) )
 
Theoremelequ2 1824 An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
 |-  ( x  =  y 
 ->  ( z  e.  x  <->  z  e.  y ) )
 
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13

The original axiom schemes of Tarski's predicate calculus are ax-4 1632, ax-5 1705, ax6v 1749, ax-7 1791, ax-8 1821, and ax-9 1823, together with rule ax-gen 1619. See http://us.metamath.org/mpeuni/mmset.html#compare 1619. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the auxiliary axiom schemes ax-10 1838, ax-11 1843, ax-12 1855, and ax-13 2000, which are not part of Tarski's axiom schemes. Each object language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-12 1855 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 2215, but they can all be proved as theorems from the above.)

Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the  x and  y in ax-6 1748 are bundled, but they are not in ax6v 1749. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1749 is the principal instance of ax-6 1748. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance  -.  A. x -.  x  =  x of ax-6 1748 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-10 1838, ax-11 1843, ax-12 1855, and ax-13 2000. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 1838, ax-11 1843, ax-12 1855, or ax-13 2000 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

The theorem schemes ax10w 1826, ax11w 1827, ax12w 1830, and ax13w 1833 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 1838, ax-11 1843, ax-12 1855, and ax-13 2000 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 1826, ax11w 1827, and ax12w 1830 is of the form  ( x  =  y  ->  ( ph  <->  ps ) ) where  ps is an auxiliary or "dummy" wff metavariable in which  x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax12wdemo 1832 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax11dgen 1828, ax12dgen 1831, ax13dgen1 1834, ax13dgen2 1835, ax13dgen3 1836, and ax13dgen4 1837. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 1838, ax-11 1843, ax-12 1855, and ax-13 2000 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.

It is interesting that Tarski used the bundled scheme ax-6 1748 in an older system, so it seems the main purpose of his later ax6v 1749 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-6 1748 as our official axiom, we show that the degenerate instance holds in ax6dgen 1825.

The case of sp 1860 is curious: originally an axiom of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form  A. x ph  ->  ph apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1808, again requiring substitution instances of  ph that meet conditions (1) and (2) above. Note that our direct proof sp 1860 requires ax-12 1855, which is not part of Tarski's system.

 
Theoremax6dgen 1825 Tarski's system uses the weaker ax6v 1749 instead of the bundled ax-6 1748, so here we show that the degenerate case of ax-6 1748 can be derived. (Contributed by NM, 23-Apr-2017.)
 |- 
 -.  A. x  -.  x  =  x
 
Theoremax10w 1826* Weak version of ax-10 1838 from which we can prove any ax-10 1838 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremax11w 1827* Weak version of ax-11 1843 from which we can prove any ax-11 1843 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1843, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax11dgen 1828 Degenerate instance of ax-11 1843 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( A. x A. x ph  ->  A. x A. x ph )
 
Theoremax12wlem 1829* Lemma for weak version of ax-12 1855. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1830. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12w 1830* Weak version of ax-12 1855 from which we can prove any ax-12 1855 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for  ph, see ax12wdemo 1832. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax12dgen 1831 Degenerate instance of ax-12 1855 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( x  =  x 
 ->  ( A. x ph  ->  A. x ( x  =  x  ->  ph )
 ) )
 
Theoremax12wdemo 1832* Example of an application of ax12w 1830 that results in an instance of ax-12 1855 for a contrived formula with mixed free and bound variables,  ( x  e.  y  /\  A. x
z  e.  x  /\  A. y A. z y  e.  x ), in place of  ph. The proof illustrates bound variable renaming with cbvalvw 1810 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
 |-  ( x  =  y 
 ->  ( A. y ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )  ->  A. x ( x  =  y  ->  ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )
 ) ) )
 
Theoremax13w 1833* Weak version (principal instance) of ax-13 2000. (Because  y and  z don't need to be distinct, this actually bundles the principal instance and the degenerate instance  ( -.  x  =  y  ->  ( y  =  y  ->  A. x
y  =  y ) ).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 1826, ax11w 1827, and ax12w 1830. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax13dgen1 1834 Degenerate instance of ax-13 2000 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  z  ->  A. x  x  =  z )
 )
 
Theoremax13dgen2 1835 Degenerate instance of ax-13 2000 where bundled variables  x and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  x  ->  A. x  y  =  x )
 )
 
Theoremax13dgen3 1836 Degenerate instance of ax-13 2000 where bundled variables  y and  z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  y  ->  A. x  y  =  y )
 )
 
Theoremax13dgen4 1837 Degenerate instance of ax-13 2000 where bundled variables  x,  y, and  z have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  x  ->  A. x  x  =  x )
 )
 
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)

In this section we introduce four additional schemes ax-10 1838, ax-11 1843, ax-12 1855, and ax-13 2000 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs and set metavariables ranging over object-language individual variables.

To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 1826, ax11w 1827, ax12w 1830, and ax13w 1833, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all set metavariables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.

An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 1855 from all others has been shown, and independence of Tarski's ax-6 1748 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.

 
1.5.1  Axiom scheme ax-10 (Quantified Negation)
 
Axiomax-10 1838 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1826) but is used as an auxiliary axiom to achieve metalogical completeness. It means that  x is not free in  -.  A. x ph. (Contributed by NM, 21-May-2008.) Use its alias hbn1 1839 instead. (New usage is discouraged.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbn1 1839 Alias for ax-10 1838 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
 |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremhbe1 1840  x is not free in  E. x ph. (Contributed by NM, 24-Jan-1993.)
 |-  ( E. x ph  ->  A. x E. x ph )
 
Theoremnfe1 1841  x is not free in  E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x E. x ph
 
Theoremmodal-5 1842 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( -.  A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
 
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
 
Axiomax-11 1843 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 1827) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremalcoms 1844 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremhbal 1845 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 12-Mar-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremalcom 1846 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theoremalrot3 1847 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y A. z ph  <->  A. y A. z A. x ph )
 
Theoremalrot4 1848 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  ( A. x A. y A. z A. w ph  <->  A. z A. w A. x A. y ph )
 
Theoremhbald 1849 Deduction form of bound-variable hypothesis builder hbal 1845. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( A. y ps  ->  A. x A. y ps ) )
 
Theoremexcom 1850 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-12 1855, ax-10 1838, ax-6 1748, ax-7 1791 and ax-5 1705. (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theoremexcomim 1851 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-12 1855, ax-10 1838, ax-6 1748, ax-7 1791 and ax-5 1705. (Revised by Wolf Lammen, 8-Jan-2018.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
Theoremexcom13 1852 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
 
Theoremexrot3 1853 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4 1854 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ph  <->  E. z E. w E. x E. y ph )
 
1.5.3  Axiom scheme ax-12 (Substitution)
 
Axiomax-12 1855 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent  A. x ( x  =  y  ->  ph ) is a way of expressing " y substituted for  x in wff  ph " (cf. sb6 2174). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-c15 2221 and was replaced with this shorter ax-12 1855 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2086. Conversely, this axiom is proved from ax-c15 2221 as theorem ax12 2235.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 2221) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 1856 and ax12v2 2084 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1830) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.)

 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax12v 1856* This is a version of ax-12 1855 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax12v2 2084 for the rederivation of ax-c15 2221 from this theorem. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1838 and ax-13 2000. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theoremax12vOLD 1857* Obsolete proof of ax12v 1856 as of 8-Dec-2019. (Contributed by Jim Kingdon, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  ->  A. x ( x  =  y  -> 
 ph ) ) )
 
Theorem19.8a 1858 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1754 for a version requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 1860. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
 |-  ( ph  ->  E. x ph )
 
Theorem19.8aOLD 1859 Obsolete proof of 19.8a 1858 as of 8-Dec-2019. (Contributed by NM, 9-Jan-1993.) (Revised by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ph )
 
Theoremsp 1860 Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint).

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2095.

This theorem shows that our obsolete axiom ax-c5 2215 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 1855. It is thought the best we can do using only Tarski's axioms is spw 1808. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

 |-  ( A. x ph  -> 
 ph )
 
Theoremaxc4 1861 Show that the original axiom ax-c4 2216 can be derived from ax-4 1632 and others. See ax4 2226 for the rederivation of ax-4 1632 from ax-c4 2216.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

 |-  ( A. x (
 A. x ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremaxc7 1862 Show that the original axiom ax-c7 2217 can be derived from ax-10 1838 and others. See ax10 2227 for the rederivation of ax-10 1838 from ax-c7 2217.

Normally, axc7 1862 should be used rather than ax-c7 2217, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremaxc7e 1863 Abbreviated version of axc7 1862. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x A. x ph  ->  ph )
 
Theoremmodal-b 1864 The analog in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
 |-  ( ph  ->  A. x  -.  A. x  -.  ph )
 
Theoremspi 1865 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
 |- 
 A. x ph   =>    |-  ph
 
Theoremsps 1866 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem2sp 1867 A double specialization (see sp 1860). Another double specialization, closer to PM*11.1, is 2stdpc4 2096. (Contributed by BJ, 15-Sep-2018.)
 |-  ( A. x A. y ph  ->  ph )
 
Theoremspsd 1868 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theorem19.2g 1869 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1752 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
 |-  ( A. x ph  ->  E. y ph )
 
Theorem19.21bi 1870 Inference form of 19.21 1906 and also deduction form of sp 1860. (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.21bbi 1871 Inference removing double quantifier. Version of 19.21bi 1870 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.23bi 1872 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 1911. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremnexr 1873 Inference form of 19.8a 1858. (Contributed by Jeff Hankins, 26-Jul-2009.)
 |- 
 -.  E. x ph   =>    |- 
 -.  ph
 
Theoremnfr 1874 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 |-  ( F/ x ph  ->  ( ph  ->  A. x ph ) )
 
Theoremnfri 1875 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfrd 1876 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremalimd 1877 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1633. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalrimi 1878 Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1906. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremnfd 1879 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremnfdh 1880 Deduce that  x is not free in  ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  F/ x ps )
 
Theoremalrimdd 1881 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1906. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremalrimd 1882 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1906. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximd 1883 Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1655. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremnexd 1884 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremnexdv 1885* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbid 1886 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbid 1887 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremnfbidf 1888 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( F/ x ps  <->  F/ x ch )
 )
 
Theorem19.3 1889 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1756 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x ph  <->  ph )
 
Theorem19.9ht 1890 A closed version of 19.9 1894. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( E. x ph  -> 
 ph ) )
 
Theorem19.9t 1891 A closed version of 19.9 1894. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
 
Theorem19.9h 1892 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theorem19.9d 1893 A deduction version of one direction of 19.9 1894. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |-  ( ps  ->  F/ x ph )   =>    |-  ( ps  ->  ( E. x ph  ->  ph )
 )
 
Theorem19.9 1894 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1755 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |-  ( E. x ph  <->  ph )
 
Theoremhbnt 1895 Closed theorem version of bound-variable hypothesis builder hbn 1896. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
 |-  ( A. x (
 ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph )
 )
 
Theoremhbn 1896 If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremhba1 1897  x is not free in  A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremnfa1 1898  x is not free in  A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x A. x ph
 
Theoremaxc4i 1899 Inference version of axc4 1861. (Contributed by NM, 3-Jan-1993.)
 |-  ( A. x ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theoremnfnf1 1900  x is not free in  F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x F/ x ph
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