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Type | Label | Description |
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Statement | ||
Theorem | gruun 10801 | A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ π΅ β π) β (π΄ βͺ π΅) β π) | ||
Theorem | gruxp 10802 | A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ π΅ β π) β (π΄ Γ π΅) β π) | ||
Theorem | grumap 10803 | A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ π΅ β π) β (π΄ βm π΅) β π) | ||
Theorem | gruixp 10804* | A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ βπ₯ β π΄ π΅ β π) β Xπ₯ β π΄ π΅ β π) | ||
Theorem | gruiin 10805* | A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ βπ₯ β π΄ π΅ β π) β β© π₯ β π΄ π΅ β π) | ||
Theorem | gruf 10806 | A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ πΉ:π΄βΆπ) β πΉ β π) | ||
Theorem | gruen 10807 | A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π β Univ β§ π΄ β π β§ (π΅ β π β§ π΅ β π΄)) β π΄ β π) | ||
Theorem | gruwun 10808 | A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ ((π β Univ β§ π β β ) β π β WUni) | ||
Theorem | intgru 10809 | The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ ((π΄ β Univ β§ π΄ β β ) β β© π΄ β Univ) | ||
Theorem | ingru 10810* | The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((Tr π΄ β§ βπ₯ β π΄ (π« π₯ β π΄ β§ βπ¦ β π΄ {π₯, π¦} β π΄ β§ βπ¦(π¦:π₯βΆπ΄ β βͺ ran π¦ β π΄))) β (π β Univ β (π β© π΄) β Univ)) | ||
Theorem | wfgru 10811 | The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
β’ (π β Univ β (π β© βͺ (π 1 β On)) β Univ) | ||
Theorem | grudomon 10812 | Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((π β Univ β§ π΄ β On β§ (π΅ β π β§ π΄ βΌ π΅)) β π΄ β π) | ||
Theorem | gruina 10813 | If a Grothendieck universe π is nonempty, then the height of the ordinals in π is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.) |
β’ π΄ = (π β© On) β β’ ((π β Univ β§ π β β ) β π΄ β Inacc) | ||
Theorem | grur1a 10814 | A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.) |
β’ π΄ = (π β© On) β β’ (π β Univ β (π 1βπ΄) β π) | ||
Theorem | grur1 10815 | A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.) |
β’ π΄ = (π β© On) β β’ ((π β Univ β§ π β βͺ (π 1 β On)) β π = (π 1βπ΄)) | ||
Theorem | grutsk1 10816 | Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10778.) (Contributed by Mario Carneiro, 17-Jun-2013.) |
β’ ((π β Tarski β§ Tr π) β π β Univ) | ||
Theorem | grutsk 10817 | Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
β’ Univ = {π₯ β Tarski β£ Tr π₯} | ||
Axiom | ax-groth 10818* | The Tarski-Grothendieck Axiom. For every set π₯ there is an inaccessible cardinal π¦ such that π¦ is not in π₯. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics". This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols (see grothprim 10829). An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ (βπ€(π€ β π§ β π€ β π¦) β§ βπ€ β π¦ βπ£(π£ β π§ β π£ β π€)) β§ βπ§(π§ β π¦ β (π§ β π¦ β¨ π§ β π¦))) | ||
Theorem | axgroth5 10819* | The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ (π« π§ β π¦ β§ βπ€ β π¦ π« π§ β π€) β§ βπ§ β π« π¦(π§ β π¦ β¨ π§ β π¦)) | ||
Theorem | axgroth2 10820* | Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ (βπ€(π€ β π§ β π€ β π¦) β§ βπ€ β π¦ βπ£(π£ β π§ β π£ β π€)) β§ βπ§(π§ β π¦ β (π¦ βΌ π§ β¨ π§ β π¦))) | ||
Theorem | grothpw 10821* | Derive the Axiom of Power Sets ax-pow 5364 from the Tarski-Grothendieck axiom ax-groth 10818. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10818. Note that ax-pow 5364 is not used by the proof. (Contributed by GΓ©rard Lang, 22-Jun-2009.) (New usage is discouraged.) |
β’ βπ¦βπ§(βπ€(π€ β π§ β π€ β π₯) β π§ β π¦) | ||
Theorem | grothpwex 10822 | Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10818. Note that ax-pow 5364 is not used by the proof. Use axpweq 5349 to obtain ax-pow 5364. Use pwex 5379 or pwexg 5377 instead. (Contributed by GΓ©rard Lang, 22-Jun-2009.) (New usage is discouraged.) |
β’ π« π₯ β V | ||
Theorem | axgroth6 10823* | The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set π₯, there exists a set π¦ containing π₯, the subsets of the members of π¦, the power sets of the members of π¦, and the subsets of π¦ of cardinality less than that of π¦. (Contributed by NM, 21-Jun-2009.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ (π« π§ β π¦ β§ π« π§ β π¦) β§ βπ§ β π« π¦(π§ βΊ π¦ β π§ β π¦)) | ||
Theorem | grothomex 10824 | The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9638). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
β’ Ο β V | ||
Theorem | grothac 10825 | The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10464). This can be put in a more conventional form via ween 10030 and dfac8 10130. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10130). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
β’ dom card = V | ||
Theorem | axgroth3 10826* | Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10430 is used to derive this version. (Contributed by NM, 26-Mar-2007.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ (βπ€(π€ β π§ β π€ β π¦) β§ βπ€ β π¦ βπ£(π£ β π§ β π£ β π€)) β§ βπ§(π§ β π¦ β ((π¦ β π§) βΌ π§ β¨ π§ β π¦))) | ||
Theorem | axgroth4 10827* | Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10454 is used to derive this version. (Contributed by NM, 16-Apr-2007.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ βπ£ β π¦ βπ€(π€ β π§ β π€ β (π¦ β© π£)) β§ βπ§(π§ β π¦ β ((π¦ β π§) βΌ π§ β¨ π§ β π¦))) | ||
Theorem | grothprimlem 10828* | Lemma for grothprim 10829. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.) |
β’ ({π’, π£} β π€ β βπ(π β π€ β§ ββ(β β π β (β = π’ β¨ β = π£)))) | ||
Theorem | grothprim 10829* | The Tarski-Grothendieck Axiom ax-groth 10818 expanded into set theory primitives using 163 symbols (allowing the defined symbols β§, β¨, β, and β). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.) |
β’ βπ¦(π₯ β π¦ β§ βπ§((π§ β π¦ β βπ£(π£ β π¦ β§ βπ€(βπ’(π’ β π€ β π’ β π§) β (π€ β π¦ β§ π€ β π£)))) β§ βπ€((π€ β π§ β π€ β π¦) β (βπ£((π£ β π§ β βπ‘βπ’(βπ(π β π€ β§ ββ(β β π β (β = π£ β¨ β = π’))) β π’ = π‘)) β§ (π£ β π¦ β (π£ β π§ β¨ βπ’(π’ β π§ β§ βπ(π β π€ β§ ββ(β β π β (β = π’ β¨ β = π£))))))) β¨ π§ β π¦)))) | ||
Theorem | grothtsk 10830 | The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.) |
β’ βͺ Tarski = V | ||
Theorem | inaprc 10831 | An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.) |
β’ Inacc β V | ||
Syntax | ctskm 10832 | Extend class definition to include the map whose value is the smallest Tarski class. |
class tarskiMap | ||
Definition | df-tskm 10833* | A function that maps a set π₯ to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.) |
β’ tarskiMap = (π₯ β V β¦ β© {π¦ β Tarski β£ π₯ β π¦}) | ||
Theorem | tskmval 10834* | Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) | ||
Theorem | tskmid 10835 | The set π΄ is an element of the smallest Tarski class that contains π΄. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
β’ (π΄ β π β π΄ β (tarskiMapβπ΄)) | ||
Theorem | tskmcl 10836 | A Tarski class that contains π΄ is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
β’ (tarskiMapβπ΄) β Tarski | ||
Theorem | sstskm 10837* | Being a part of (tarskiMapβπ΄). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯))) | ||
Theorem | eltskm 10838* | Belonging to (tarskiMapβπ΄). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯))) | ||
This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 11167). After that, we derive their basic properties, various operations like addition (df-add 11121) and sine (df-sin 16013), and subsets such as the integers (df-z 12559) and natural numbers (df-nn 12213). | ||
Syntax | cnpi 10839 |
The set of positive integers, which is the set of natural numbers Ο
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 11139. The actual set of Dedekind cuts is defined by df-np 10976. |
class N | ||
Syntax | cpli 10840 | Positive integer addition. |
class +N | ||
Syntax | cmi 10841 | Positive integer multiplication. |
class Β·N | ||
Syntax | clti 10842 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 10843 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 10844 | Positive pre-fraction multiplication. |
class Β·pQ | ||
Syntax | cltpq 10845 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 10846 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 10847 | Set of positive fractions. |
class Q | ||
Syntax | c1q 10848 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cerq 10849 | Positive fraction equivalence class. |
class [Q] | ||
Syntax | cplq 10850 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 10851 | Positive fraction multiplication. |
class Β·Q | ||
Syntax | crq 10852 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 10853 | Positive fraction ordering relation. |
class <Q | ||
Syntax | cnp 10854 | Set of positive reals. |
class P | ||
Syntax | c1p 10855 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 10856 | Positive real addition. |
class +P | ||
Syntax | cmp 10857 | Positive real multiplication. |
class Β·P | ||
Syntax | cltp 10858 | Positive real ordering relation. |
class <P | ||
Syntax | cer 10859 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 10860 | Set of signed reals. |
class R | ||
Syntax | c0r 10861 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 10862 | The signed real constant 1. |
class 1R | ||
Syntax | cm1r 10863 | The signed real constant -1. |
class -1R | ||
Syntax | cplr 10864 | Signed real addition. |
class +R | ||
Syntax | cmr 10865 | Signed real multiplication. |
class Β·R | ||
Syntax | cltr 10866 | Signed real ordering relation. |
class <R | ||
Definition | df-ni 10867 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
β’ N = (Ο β {β }) | ||
Definition | df-pli 10868 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
β’ +N = ( +o βΎ (N Γ N)) | ||
Definition | df-mi 10869 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
β’ Β·N = ( Β·o βΎ (N Γ N)) | ||
Definition | df-lti 10870 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.) |
β’ <N = ( E β© (N Γ N)) | ||
Theorem | elni 10871 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
β’ (π΄ β N β (π΄ β Ο β§ π΄ β β )) | ||
Theorem | elni2 10872 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.) |
β’ (π΄ β N β (π΄ β Ο β§ β β π΄)) | ||
Theorem | pinn 10873 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
β’ (π΄ β N β π΄ β Ο) | ||
Theorem | pion 10874 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.) |
β’ (π΄ β N β π΄ β On) | ||
Theorem | piord 10875 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.) |
β’ (π΄ β N β Ord π΄) | ||
Theorem | niex 10876 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
β’ N β V | ||
Theorem | 0npi 10877 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
β’ Β¬ β β N | ||
Theorem | 1pi 10878 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.) |
β’ 1o β N | ||
Theorem | addpiord 10879 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ +N π΅) = (π΄ +o π΅)) | ||
Theorem | mulpiord 10880 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ Β·N π΅) = (π΄ Β·o π΅)) | ||
Theorem | mulidpi 10881 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
β’ (π΄ β N β (π΄ Β·N 1o) = π΄) | ||
Theorem | ltpiord 10882 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ <N π΅ β π΄ β π΅)) | ||
Theorem | ltsopi 10883 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
β’ <N Or N | ||
Theorem | ltrelpi 10884 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.) |
β’ <N β (N Γ N) | ||
Theorem | dmaddpi 10885 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
β’ dom +N = (N Γ N) | ||
Theorem | dmmulpi 10886 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
β’ dom Β·N = (N Γ N) | ||
Theorem | addclpi 10887 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ +N π΅) β N) | ||
Theorem | mulclpi 10888 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ Β·N π΅) β N) | ||
Theorem | addcompi 10889 | Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
β’ (π΄ +N π΅) = (π΅ +N π΄) | ||
Theorem | addasspi 10890 | Addition of positive integers is associative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
β’ ((π΄ +N π΅) +N πΆ) = (π΄ +N (π΅ +N πΆ)) | ||
Theorem | mulcompi 10891 | Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.) |
β’ (π΄ Β·N π΅) = (π΅ Β·N π΄) | ||
Theorem | mulasspi 10892 | Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.) |
β’ ((π΄ Β·N π΅) Β·N πΆ) = (π΄ Β·N (π΅ Β·N πΆ)) | ||
Theorem | distrpi 10893 | Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.) |
β’ (π΄ Β·N (π΅ +N πΆ)) = ((π΄ Β·N π΅) +N (π΄ Β·N πΆ)) | ||
Theorem | addcanpi 10894 | Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β ((π΄ +N π΅) = (π΄ +N πΆ) β π΅ = πΆ)) | ||
Theorem | mulcanpi 10895 | Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β ((π΄ Β·N π΅) = (π΄ Β·N πΆ) β π΅ = πΆ)) | ||
Theorem | addnidpi 10896 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.) |
β’ (π΄ β N β Β¬ (π΄ +N π΅) = π΄) | ||
Theorem | ltexpi 10897* | Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.) |
β’ ((π΄ β N β§ π΅ β N) β (π΄ <N π΅ β βπ₯ β N (π΄ +N π₯) = π΅)) | ||
Theorem | ltapi 10898 | Ordering property of addition for positive integers. (Contributed by NM, 7-Mar-1996.) (New usage is discouraged.) |
β’ (πΆ β N β (π΄ <N π΅ β (πΆ +N π΄) <N (πΆ +N π΅))) | ||
Theorem | ltmpi 10899 | Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.) |
β’ (πΆ β N β (π΄ <N π΅ β (πΆ Β·N π΄) <N (πΆ Β·N π΅))) | ||
Theorem | 1lt2pi 10900 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.) |
β’ 1o <N (1o +N 1o) |
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