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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | c3o 6301 | Extend the definition of a class to include the ordinal number 3. |
class 3o | ||
Syntax | c4o 6302 | Extend the definition of a class to include the ordinal number 4. |
class 4o | ||
Syntax | coa 6303 | Extend the definition of a class to include the ordinal addition operation. |
class +o | ||
Syntax | comu 6304 | Extend the definition of a class to include the ordinal multiplication operation. |
class ·o | ||
Syntax | coei 6305 | Extend the definition of a class to include the ordinal exponentiation operation. |
class ↑o | ||
Definition | df-1o 6306 | Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o = suc ∅ | ||
Definition | df-2o 6307 | Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) |
⊢ 2o = suc 1o | ||
Definition | df-3o 6308 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ 3o = suc 2o | ||
Definition | df-4o 6309 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ 4o = suc 3o | ||
Definition | df-oadd 6310* | 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 6311* | 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-oexpi 6312* |
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)) | ||
Theorem | 1on 6313 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o ∈ On | ||
Theorem | 1oex 6314 | Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.) |
⊢ 1o ∈ V | ||
Theorem | 2on 6315 | Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ 2o ∈ On | ||
Theorem | 2on0 6316 | Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) |
⊢ 2o ≠ ∅ | ||
Theorem | 3on 6317 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 3o ∈ On | ||
Theorem | 4on 6318 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 4o ∈ On | ||
Theorem | df1o2 6319 | Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.) |
⊢ 1o = {∅} | ||
Theorem | df2o3 6320 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 2o = {∅, 1o} | ||
Theorem | df2o2 6321 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
⊢ 2o = {∅, {∅}} | ||
Theorem | 1n0 6322 | Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
⊢ 1o ≠ ∅ | ||
Theorem | xp01disj 6323 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ | ||
Theorem | xp01disjl 6324 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) |
⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ | ||
Theorem | ordgt0ge1 6325 | Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | ||
Theorem | ordge1n0im 6326 | An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.) |
⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅)) | ||
Theorem | el1o 6327 | Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) |
⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅) | ||
Theorem | dif1o 6328 | 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 | 2oconcl 6329 | Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) | ||
Theorem | 0lt1o 6330 | Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.) |
⊢ ∅ ∈ 1o | ||
Theorem | 0lt2o 6331 | Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.) |
⊢ ∅ ∈ 2o | ||
Theorem | 1lt2o 6332 | Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.) |
⊢ 1o ∈ 2o | ||
Theorem | oafnex 6333 | The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.) |
⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V | ||
Theorem | sucinc 6334* | Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) |
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) ⇒ ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥) | ||
Theorem | sucinc2 6335* | Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) |
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧) ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) | ||
Theorem | fnoa 6336 | Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.) |
⊢ +o Fn (On × On) | ||
Theorem | oaexg 6337 | Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +o 𝐵) ∈ V) | ||
Theorem | omfnex 6338* | The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V) | ||
Theorem | fnom 6339 | Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.) |
⊢ ·o Fn (On × On) | ||
Theorem | omexg 6340 | Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ·o 𝐵) ∈ V) | ||
Theorem | fnoei 6341 | Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
⊢ ↑o Fn (On × On) | ||
Theorem | oeiexg 6342 | Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑o 𝐵) ∈ V) | ||
Theorem | oav 6343* | 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 6344* | 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 | oeiv 6345* | Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) | ||
Theorem | oa0 6346 | 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 6347 | 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 ∅) = ∅) | ||
Theorem | oei0 6348 | 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 | oacl 6349 | 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) | ||
Theorem | omcl 6350 | 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) | ||
Theorem | oeicl 6351 | Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | ||
Theorem | oav2 6352* | Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥))) | ||
Theorem | oasuc 6353 | 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 | omv2 6354* | Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) | ||
Theorem | onasuc 6355 | 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)) | ||
Theorem | oa1suc 6356 | 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 | o1p1e2 6357 | 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) |
⊢ (1o +o 1o) = 2o | ||
Theorem | oawordi 6358 | Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) | ||
Theorem | oawordriexmid 6359* | A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6358. (Contributed by Jim Kingdon, 15-May-2022.) |
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | oaword1 6360 | 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)) | ||
Theorem | omsuc 6361 | 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 | onmsuc 6362 | 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 | nna0 6363 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) | ||
Theorem | nnm0 6364 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | ||
Theorem | nnasuc 6365 | 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)) | ||
Theorem | nnmsuc 6366 | 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)) | ||
Theorem | nna0r 6367 | 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) | ||
Theorem | nnm0r 6368 | 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) = ∅) | ||
Theorem | nnacl 6369 | 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) | ||
Theorem | nnmcl 6370 | 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) | ||
Theorem | nnacli 6371 | ω is closed under addition. Inference form of nnacl 6369. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 +o 𝐵) ∈ ω | ||
Theorem | nnmcli 6372 | ω is closed under multiplication. Inference form of nnmcl 6370. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 ·o 𝐵) ∈ ω | ||
Theorem | nnacom 6373 | 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)) | ||
Theorem | nnaass 6374 | 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))) | ||
Theorem | nndi 6375 | 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))) | ||
Theorem | nnmass 6376 | 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))) | ||
Theorem | nnmsucr 6377 | 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)) | ||
Theorem | nnmcom 6378 | 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)) | ||
Theorem | nndir 6379 | 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))) | ||
Theorem | nnsucelsuc 6380 | Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4419, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4440. (Contributed by Jim Kingdon, 25-Aug-2019.) |
⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) | ||
Theorem | nnsucsssuc 6381 | Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4420, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4437. (Contributed by Jim Kingdon, 25-Aug-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) | ||
Theorem | nntri3or 6382 | Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
Theorem | nntri2 6383 | A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | ||
Theorem | nnsucuniel 6384 | Given an element 𝐴 of the union of a natural number 𝐵, suc 𝐴 is an element of 𝐵 itself. The reverse direction holds for all ordinals (sucunielr 4421). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4441). (Contributed by Jim Kingdon, 13-Mar-2022.) |
⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)) | ||
Theorem | nntri1 6385 | A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | ||
Theorem | nntri3 6386 | A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))) | ||
Theorem | nntri2or2 6387 | A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
Theorem | nndceq 6388 | Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4526. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵) | ||
Theorem | nndcel 6389 | Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵) | ||
Theorem | nnsseleq 6390 | For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | nnsssuc 6391 | A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | ||
Theorem | nntr2 6392 | Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.) |
⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | dcdifsnid 6393* | If we remove a single element from a set with decidable equality then put it back in, we end up with the original set. This strengthens difsnss 3661 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.) |
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) | ||
Theorem | fnsnsplitdc 6394* | Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.) |
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉})) | ||
Theorem | funresdfunsndc 6395* | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) = 𝐹) | ||
Theorem | nndifsnid 6396 | 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 3661 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) | ||
Theorem | nnaordi 6397 | 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) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | ||
Theorem | nnaord 6398 | 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) → (𝐴 ∈ 𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | ||
Theorem | nnaordr 6399 | Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))) | ||
Theorem | nnaword 6400 | Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) |
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