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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordge1n0 6701 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)

Theoremel1o 6702 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Theoremdif1o 6703 Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.)

Theoremondif1 6704 Two ways to say that is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)

Theoremondif2 6705 Two ways to say that is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)

Theorem2oconcl 6706 Closure of the pair swapping function on . (Contributed by Mario Carneiro, 27-Sep-2015.)

Theorem0lt1o 6707 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)

Theoremdif20el 6708 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)

Theorem0we1 6709 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorembrwitnlem 6710 Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremfnoa 6711 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnom 6712 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnoe 6713 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoav 6714* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomv 6715* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoe0lem 6716 A helper lemma for oe0 6725 and others. (Contributed by NM, 6-Jan-2005.)

Theoremoev 6717* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoevn0 6718* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoa0 6719 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0 6720 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoe0m 6721 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0x 6722 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6720, this version works whether or not is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.)

Theoremoe0m0 6723 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)

Theoremoe0m1 6724 Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)

Theoremoe0 6725 Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoev2 6726* Alternate value of ordinal exponentiation. Compare oev 6717. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoasuc 6727 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuclem 6728* Lemma for oesuc 6730. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsuc 6729 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuc 6730 Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremonasuc 6731 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6727 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremonmsuc 6732 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremonesuc 6733 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremoa1suc 6734 Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremoalim 6735* Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomlim 6736* Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoelim 6737* Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoacl 6738 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremomcl 6739 Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoecl 6740 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoa0r 6741 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremom0r 6742 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremo1p1e2 6743 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)

Theoremom1 6744 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)

Theoremom1r 6745 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremoe1 6746 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)

Theoremoe1m 6747 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Theoremoaordi 6748 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaord 6749 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)

Theoremoacan 6750 Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaword 6751 Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoawordri 6752 Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)

Theoremoaord1 6753 An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of [Suppes] p. 209 and its converse. (Contributed by NM, 6-Dec-2004.)

Theoremoaword1 6754 An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (For the other part see oaord1 6753.) (Contributed by NM, 6-Dec-2004.)

Theoremoaword2 6755 An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.)

Theoremoawordeulem 6756* Lemma for oawordex 6759. (Contributed by NM, 11-Dec-2004.)

Theoremoawordeu 6757* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)

Theoremoawordexr 6758* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)

Theoremoawordex 6759* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6757 for uniqueness. (Contributed by NM, 12-Dec-2004.)

Theoremoaordex 6760* Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse. (Contributed by NM, 12-Dec-2004.)

Theoremoa00 6761 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)

Theoremoalimcl 6762 The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)

Theoremoaass 6763 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)

Theoremoarec 6764* Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)

Theoremoaf1o 6765* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1olem 6766* Lemma for oacomf1o 6767. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1o 6767* Define a bijection from to . Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 7562). (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomordi 6768 Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomord2 6769 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)

Theoremomord 6770 Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomcan 6771 Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)

Theoremomword 6772 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordi 6773 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordri 6774 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)

Theoremomword1 6775 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremomword2 6776 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremom00 6777 The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.)

Theoremom00el 6778 The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)

Theoremomordlim 6779* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Theoremomlimcl 6780 The product of any nonzero ordinal with a limit ordinal is a limit ordinal. Proposition 8.24 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Theoremodi 6781 Distributive law for ordinal arithmetic. Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)

Theoremomass 6782 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)

Theoremoneo 6783 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)

Theoremomeulem1 6784* Lemma for omeu 6787: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremomeulem2 6785 Lemma for omeu 6787: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Theoremomopth2 6786 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomeu 6787* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoen0 6788 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)

Theoremoeordi 6789 Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeord 6790 Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoecan 6791 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeword 6792 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoewordi 6793 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Theoremoewordri 6794 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)

Theoremoeworde 6795 Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeordsuc 6796 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)

Theoremoelim2 6797* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)

Theoremoeoalem 6798 Lemma for oeoa 6799. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoa 6799 Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoelem 6800 Lemma for oeoe 6801. (Contributed by Eric Schmidt, 26-May-2009.)

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