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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-1o 6001 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1𝑜 = suc ∅

Definitiondf-2o 6002 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2𝑜 = suc 1𝑜

Definitiondf-3o 6003 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3𝑜 = suc 2𝑜

Definitiondf-4o 6004 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4𝑜 = suc 3𝑜

+𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

Definitiondf-omul 6006* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))

Definitiondf-oexpi 6007* Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑𝑜 𝐴 to be 1𝑜 for all 𝐴 ∈ On, in order to avoid having different cases for whether the base is or not. (Contributed by Mario Carneiro, 4-Jul-2019.)

𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))

Theorem1on 6008 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ On

Theorem2on 6009 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2𝑜 ∈ On

Theorem2on0 6010 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2𝑜 ≠ ∅

Theorem3on 6011 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ On

Theorem4on 6012 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ On

Theoremdf1o2 6013 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1𝑜 = {∅}

Theoremdf2o3 6014 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2𝑜 = {∅, 1𝑜}

Theoremdf2o2 6015 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2𝑜 = {∅, {∅}}

Theorem1n0 6016 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1𝑜 ≠ ∅

Theoremxp01disj 6017 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Theoremordgt0ge1 6018 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))

Theoremordge1n0im 6019 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
(Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))

Theoremel1o 6020 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1𝑜𝐴 = ∅)

Theoremdif1o 6021 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 ≠ ∅))

Theorem2oconcl 6022 Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)

Theorem0lt1o 6023 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1𝑜

Theoremoafnex 6024 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
(𝑥 ∈ V ↦ suc 𝑥) Fn V

Theoremsucinc 6025* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       𝑥 𝑥 ⊆ (𝐹𝑥)

Theoremsucinc2 6026* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremfnoa 6027 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
+𝑜 Fn (On × On)

Theoremoaexg 6028 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 +𝑜 𝐵) ∈ V)

Theoremomfnex 6029* The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
(𝐴𝑉 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V)

Theoremfnom 6030 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
·𝑜 Fn (On × On)

Theoremomexg 6031 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ·𝑜 𝐵) ∈ V)

Theoremfnoei 6032 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
𝑜 Fn (On × On)

Theoremoeiexg 6033 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑜 𝐵) ∈ V)

Theoremoav 6034* 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 6035* 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))

Theoremoeiv 6036* Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))

Theoremoa0 6037 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 6038 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)

Theoremoei0 6039 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)

Theoremoacl 6040 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)

Theoremomcl 6041 Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)

Theoremoeicl 6042 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)

Theoremoav2 6043* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +𝑜 𝑥)))

Theoremoasuc 6044 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))

Theoremomv2 6045* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))

Theoremonasuc 6046 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremoa1suc 6047 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)

Theoremo1p1e2 6048 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1𝑜 +𝑜 1𝑜) = 2𝑜

Theoremoawordi 6049 Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Theoremoaword1 6050 An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))

Theoremomsuc 6051 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))

Theoremonmsuc 6052 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))

2.6.23  Natural number arithmetic

Theoremnna0 6053 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)

Theoremnnm0 6054 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)

Theoremnnasuc 6055 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremnnmsuc 6056 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Theoremnna0r 6057 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
(𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)

Theoremnnm0r 6058 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)

Theoremnnacl 6059 Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)

Theoremnnmcl 6060 Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)

Theoremnnacli 6061 ω is closed under addition. Inference form of nnacl 6059. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +𝑜 𝐵) ∈ ω

Theoremnnmcli 6062 ω is closed under multiplication. Inference form of nnmcl 6060. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·𝑜 𝐵) ∈ ω

Theoremnnacom 6063 Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))

Theoremnnaass 6064 Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Theoremnndi 6065 Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))

Theoremnnmass 6066 Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))

Theoremnnmsucr 6067 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))

Theoremnnmcom 6068 Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))

Theoremnndir 6069 Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))

Theoremnnsucelsuc 6070 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4234, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4255. (Contributed by Jim Kingdon, 25-Aug-2019.)
(𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Theoremnnsucsssuc 6071 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4235, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4252. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Theoremnntri3or 6072 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremnntri2 6073 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Theoremnntri1 6074 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremnntri3 6075 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))

Theoremnntri2or2 6076 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))

Theoremnndceq 6077 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4339. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)

Theoremnndcel 6078 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)

Theoremnnsseleq 6079 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremnndifsnid 6080 If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3510 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Theoremnnaordi 6081 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaord 6082 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaordr 6083 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))

Theoremnnaword 6084 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Theoremnnacan 6085 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶))

Theoremnnaword1 6086 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))

Theoremnnaword2 6087 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴))

Theoremnnawordi 6088 Adding to both sides of an inequality in ω (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Theoremnnmordi 6089 Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmord 6090 Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmword 6091 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
(((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Theoremnnmcan 6092 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶))

Theorem1onn 6093 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ ω

Theorem2onn 6094 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2𝑜 ∈ ω

Theorem3onn 6095 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ ω

Theorem4onn 6096 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ ω

Theoremnnm1 6097 Multiply an element of ω by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴)

Theoremnnm2 6098 Multiply an element of ω by 2𝑜 (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴))

Theoremnn2m 6099 Multiply an element of ω by 2𝑜 (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (2𝑜 ·𝑜 𝐴) = (𝐴 +𝑜 𝐴))

Theoremnnaordex 6100* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))

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