P. 82 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Syntax	coe 8101	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 8102	Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
⊢ 1o = suc ∅
Definition	df-2o 8103	Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
⊢ 2o = suc 1o
Definition	df-3o 8104	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 3o = suc 2o
Definition	df-4o 8105	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 4o = suc 3o
Definition	df-oadd 8106*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
Definition	df-omul 8107*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Definition	df-oexp 8108*	Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
Theorem	1on 8109	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ On
Theorem	1oex 8110	Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.)
⊢ 1o ∈ V
Theorem	2on 8111	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ 2o ∈ On
Theorem	2oex 8112	2o is a set. (Contributed by BJ, 6-Apr-2019.)
⊢ 2o ∈ V
Theorem	2on0 8113	Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ 2o ≠ ∅
Theorem	3on 8114	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ On
Theorem	4on 8115	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ On
Theorem	df1o2 8116	Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
⊢ 1o = {∅}
Theorem	df2o3 8117	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 2o = {∅, 1o}
Theorem	df2o2 8118	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
⊢ 2o = {∅, {∅}}
Theorem	1n0 8119	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
⊢ 1o ≠ ∅
Theorem	xp01disj 8120	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Theorem	xp01disjl 8121	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Theorem	ordgt0ge1 8122	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
Theorem	ordge1n0 8123	An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	el1o 8124	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
Theorem	dif1o 8125	Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
Theorem	ondif1 8126	Two ways to say that 𝐴 is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
Theorem	ondif2 8127	Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
Theorem	2oconcl 8128	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)
Theorem	0lt1o 8129	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ ∅ ∈ 1o
Theorem	dif20el 8130	An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
Theorem	0we1 8131	The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ ∅ We 1o
Theorem	brwitnlem 8132	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) ≠ ∅)
Theorem	fnoa 8133	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ +o Fn (On × On)
Theorem	fnom 8134	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ·o Fn (On × On)
Theorem	fnoe 8135	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
⊢ ↑o Fn (On × On)
Theorem	oav 8136*	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))
Theorem	omv 8137*	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))
Theorem	oe0lem 8138	A helper lemma for oe0 8147 and others. (Contributed by NM, 6-Jan-2005.)
⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝜓)    &   ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Theorem	oev 8139*	Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Theorem	oevn0 8140*	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) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
Theorem	oa0 8141	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)
Theorem	om0 8142	Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. See om0x 8144 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 ∅) = ∅)
Theorem	oe0m 8143	Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
Theorem	om0x 8144	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8142, 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 ∅) = ∅
Theorem	oe0m0 8145	Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
⊢ (∅ ↑o ∅) = 1o
Theorem	oe0m1 8146	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) = ∅))
Theorem	oe0 8147	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 → (𝐴 ↑o ∅) = 1o)
Theorem	oev2 8148*	Alternate value of ordinal exponentiation. Compare oev 8139. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
Theorem	oasuc 8149	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))
Theorem	oesuclem 8150*	Lemma for oesuc 8152. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ Lim 𝑋    &   ⊢ (𝐵 ∈ 𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	omsuc 8151	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))
Theorem	oesuc 8152	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) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	onasuc 8153	Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8149 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Theorem	onmsuc 8154	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))
Theorem	onesuc 8155	Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	oa1suc 8156	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)
Theorem	oalim 8157*	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 ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
Theorem	omlim 8158*	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 ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
Theorem	oelim 8159*	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 ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))
Theorem	oacl 8160	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)
Theorem	omcl 8161	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)
Theorem	oecl 8162	Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
Theorem	oa0r 8163	Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Theorem	om0r 8164	Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Theorem	o1p1e2 8165	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
⊢ (1o +o 1o) = 2o
Theorem	o2p2e4 8166	2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6197. For the usual proof using complex numbers, see 2p2e4 11773. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5190, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
⊢ (2o +o 2o) = 4o
Theorem	o2p2e4OLD 8167	Obsolete version of o2p2e4 8166 as of 23-Mar-2024. (Contributed by NM, 18-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (2o +o 2o) = 4o
Theorem	om1 8168	Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)
⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
Theorem	om1r 8169	Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Theorem	oe1 8170	Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴)
Theorem	oe1m 8171	Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
Theorem	oaordi 8172	Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Theorem	oaord 8173	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) → (𝐴 ∈ 𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Theorem	oacan 8174	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))
Theorem	oaword 8175	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) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
Theorem	oawordri 8176	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) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
Theorem	oaord1 8177	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) → (∅ ∈ 𝐵 ↔ 𝐴 ∈ (𝐴 +o 𝐵)))
Theorem	oaword1 8178	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 8177.) (Contributed by NM, 6-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
Theorem	oaword2 8179	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))
Theorem	oawordeulem 8180*	Lemma for oawordex 8183. (Contributed by NM, 11-Dec-2004.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    &   ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}    ⇒   ⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Theorem	oawordeu 8181*	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) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Theorem	oawordexr 8182*	Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) → 𝐴 ⊆ 𝐵)
Theorem	oawordex 8183*	Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 8181 for uniqueness. (Contributed by NM, 12-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
Theorem	oaordex 8184*	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) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Theorem	oa00 8185	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 = ∅)))
Theorem	oalimcl 8186	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 ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))
Theorem	oaass 8187	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)))
Theorem	oarec 8188*	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 (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
Theorem	oaf1o 8189*	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))
Theorem	oacomf1olem 8190*	Lemma for oacomf1o 8191. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅))
Theorem	oacomf1o 8191*	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 9114). (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))
Theorem	omordi 8192	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) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	omord2 8193	Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	omord 8194	Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	omcan 8195	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))
Theorem	omword 8196	Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
Theorem	omwordi 8197	Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
Theorem	omwordri 8198	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) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
Theorem	omword1 8199	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))
Theorem	omword2 8200	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))