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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordge1n0 8101 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))

Theoremel1o 8102 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1o𝐴 = ∅)

Theoremdif1o 8103 Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴𝐵𝐴 ≠ ∅))

Theoremondif1 8104 Two ways to say that 𝐴 is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))

Theoremondif2 8105 Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))

Theorem2oconcl 8106 Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2o → (1o𝐴) ∈ 2o)

Theorem0lt1o 8107 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1o

Theoremdif20el 8108 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
(𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)

Theorem0we1 8109 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
∅ We 1o

Theorembrwitnlem 8110 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.)
𝑅 = (𝑂 “ (V ∖ 1o))    &   𝑂 Fn 𝑋       (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

Theoremfnoa 8111 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
+o Fn (On × On)

Theoremfnom 8112 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
·o Fn (On × On)

Theoremfnoe 8113 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
o Fn (On × On)

Theoremoav 8114* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Theoremomv 8115* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Theoremoe0lem 8116 A helper lemma for oe0 8125 and others. (Contributed by NM, 6-Jan-2005.)
((𝜑𝐴 = ∅) → 𝜓)    &   (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)       ((𝐴 ∈ On ∧ 𝜑) → 𝜓)

Theoremoev 8117* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))

Theoremoevn0 8118* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

Theoremoa0 8119 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)

Theoremom0 8120 Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. See om0x 8122 for a way to remove the antecedent 𝐴 ∈ On. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 ·o ∅) = ∅)

Theoremoe0m 8121 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (∅ ↑o 𝐴) = (1o𝐴))

Theoremom0x 8122 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8120, this version works whether or not 𝐴 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.) (New usage is discouraged.)
(𝐴 ·o ∅) = ∅

Theoremoe0m0 8123 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
(∅ ↑o ∅) = 1o

Theoremoe0m1 8124 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.)
(𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅))

Theoremoe0 8125 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.)
(𝐴 ∈ On → (𝐴o ∅) = 1o)

Theoremoev2 8126* Alternate value of ordinal exponentiation. Compare oev 8117. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))

Theoremoasuc 8127 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Theoremoesuclem 8128* Lemma for oesuc 8130. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Lim 𝑋    &   (𝐵𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))       ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Theoremomsuc 8129 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Theoremoesuc 8130 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Theoremonasuc 8131 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8127 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Theoremonmsuc 8132 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))

Theoremonesuc 8133 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Theoremoa1suc 8134 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.)
(𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)

Theoremoalim 8135* 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.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))

Theoremomlim 8136* 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.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥))

Theoremoelim 8137* 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.)
(((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥))

Theoremoacl 8138 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

Theoremomcl 8139 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Theoremoecl 8140 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)

Theoremoa0r 8141 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
(𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Theoremom0r 8142 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
(𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

Theoremo1p1e2 8143 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1o +o 1o) = 2o

Theoremo2p2e4 8144 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6173. For the usual proof using complex numbers, see 2p2e4 11751. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5166, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
(2o +o 2o) = 4o

Theoremo2p2e4OLD 8145 Obsolete version of o2p2e4 8144 as of 23-Mar-2024. (Contributed by NM, 18-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(2o +o 2o) = 4o

Theoremom1 8146 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)
(𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)

Theoremom1r 8147 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
(𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)

Theoremoe1 8148 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
(𝐴 ∈ On → (𝐴o 1o) = 𝐴)

Theoremoe1m 8149 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
(𝐴 ∈ On → (1oo 𝐴) = 1o)

Theoremoaordi 8150 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Theoremoaord 8151 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Theoremoacan 8152 Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ 𝐵 = 𝐶))

Theoremoaword 8153 Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))

Theoremoawordri 8154 Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))

Theoremoaord1 8155 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ∈ (𝐴 +o 𝐵)))

Theoremoaword1 8156 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 8155.) (Contributed by NM, 6-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))

Theoremoaword2 8157 An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))

Theoremoawordeulem 8158* Lemma for oawordex 8161. (Contributed by NM, 11-Dec-2004.)
𝐴 ∈ On    &   𝐵 ∈ On    &   𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}       (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Theoremoawordeu 8159* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Theoremoawordexr 8160* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵)

Theoremoawordex 8161* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 8159 for uniqueness. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))

Theoremoaordex 8162* 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Theoremoa00 8163 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Theoremoalimcl 8164 The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))

Theoremoaass 8165 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Theoremoarec 8166* Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))

Theoremoaf1o 8167* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)
(𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴))

Theoremoacomf1olem 8168* Lemma for oacomf1o 8169. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥𝐴 ↦ (𝐵 +o 𝑥))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))

Theoremoacomf1o 8169* Define a bijection from 𝐴 +o 𝐵 to 𝐵 +o 𝐴. Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g., oancom 9092). (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = ((𝑥𝐴 ↦ (𝐵 +o 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +o 𝑥)))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))

Theoremomordi 8170 Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
(((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Theoremomord2 8171 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Theoremomord 8172 Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Theoremomcan 8173 Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Theoremomword 8174 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Theoremomwordi 8175 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Theoremomwordri 8176 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Theoremomword1 8177 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))

Theoremomword2 8178 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Theoremom00 8179 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

Theoremom00el 8180 The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))

Theoremomordlim 8181* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)
(((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) ∧ 𝐶 ∈ (𝐴 ·o 𝐵)) → ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥))

Theoremomlimcl 8182 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.)
(((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

Theoremodi 8183 Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Theoremomass 8184 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

Theoremoneo 8185 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Theoremomeulem1 8186* Lemma for omeu 8189: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)

Theoremomeulem2 8187 Lemma for omeu 8189: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Theoremomopth2 8188 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))

Theoremomeu 8189* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))

Theoremoen0 8190 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))

Theoremoeordi 8191 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.)
((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))

Theoremoeord 8192 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))

Theoremoecan 8193 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))

Theoremoeword 8194 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Theoremoewordi 8195 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Theoremoewordri 8196 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))

Theoremoeworde 8197 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.)
((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))

Theoremoeordsuc 8198 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))

Theoremoelim2 8199* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑥 ∈ (𝐵 ∖ 1o)(𝐴o 𝑥))

Theoremoeoalem 8200 Lemma for oeoa 8201. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴    &   𝐵 ∈ On       (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))

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