HomeHome Intuitionistic Logic Explorer
Theorem List (p. 64 of 133)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxc3o 6301 Extend the definition of a class to include the ordinal number 3.
class 3o
 
Syntaxc4o 6302 Extend the definition of a class to include the ordinal number 4.
class 4o
 
Syntaxcoa 6303 Extend the definition of a class to include the ordinal addition operation.
class +o
 
Syntaxcomu 6304 Extend the definition of a class to include the ordinal multiplication operation.
class ·o
 
Syntaxcoei 6305 Extend the definition of a class to include the ordinal exponentiation operation.
class o
 
Definitiondf-1o 6306 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1o = suc ∅
 
Definitiondf-2o 6307 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2o = suc 1o
 
Definitiondf-3o 6308 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3o = suc 2o
 
Definitiondf-4o 6309 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4o = suc 3o
 
Definitiondf-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))
 
Definitiondf-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))
 
Definitiondf-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))
 
Theorem1on 6313 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1o ∈ On
 
Theorem1oex 6314 Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
1o ∈ V
 
Theorem2on 6315 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2o ∈ On
 
Theorem2on0 6316 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2o ≠ ∅
 
Theorem3on 6317 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ On
 
Theorem4on 6318 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ On
 
Theoremdf1o2 6319 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1o = {∅}
 
Theoremdf2o3 6320 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2o = {∅, 1o}
 
Theoremdf2o2 6321 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2o = {∅, {∅}}
 
Theorem1n0 6322 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1o ≠ ∅
 
Theoremxp01disj 6323 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
 
Theoremxp01disjl 6324 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
(({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
 
Theoremordgt0ge1 6325 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
 
Theoremordge1n0im 6326 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
(Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
 
Theoremel1o 6327 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1o𝐴 = ∅)
 
Theoremdif1o 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) ↔ (𝐴𝐵𝐴 ≠ ∅))
 
Theorem2oconcl 6329 Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
 
Theorem0lt1o 6330 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1o
 
Theorem0lt2o 6331 Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
∅ ∈ 2o
 
Theorem1lt2o 6332 Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
1o ∈ 2o
 
Theoremoafnex 6333 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
(𝑥 ∈ V ↦ suc 𝑥) Fn V
 
Theoremsucinc 6334* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       𝑥 𝑥 ⊆ (𝐹𝑥)
 
Theoremsucinc2 6335* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremfnoa 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)
 
Theoremoaexg 6337 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 +o 𝐵) ∈ V)
 
Theoremomfnex 6338* 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 6339 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
·o Fn (On × On)
 
Theoremomexg 6340 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ·o 𝐵) ∈ V)
 
Theoremfnoei 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)
 
Theoremoeiexg 6342 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴o 𝐵) ∈ V)
 
Theoremoav 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))
 
Theoremomv 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))
 
Theoremoeiv 6345* 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 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)
 
Theoremom0 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 ∅) = ∅)
 
Theoremoei0 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 → (𝐴o ∅) = 1o)
 
Theoremoacl 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)
 
Theoremomcl 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)
 
Theoremoeicl 6351 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
 
Theoremoav2 6352* 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 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))
 
Theoremomv2 6354* 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 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))
 
Theoremoa1suc 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)
 
Theoremo1p1e2 6357 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1o +o 1o) = 2o
 
Theoremoawordi 6358 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)))
 
Theoremoawordriexmid 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) → (𝑎𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))       (𝜑 ∨ ¬ 𝜑)
 
Theoremoaword1 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))
 
Theoremomsuc 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))
 
Theoremonmsuc 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))
 
2.6.23  Natural number arithmetic
 
Theoremnna0 6363 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
 
Theoremnnm0 6364 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
 
Theoremnnasuc 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))
 
Theoremnnmsuc 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))
 
Theoremnna0r 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)
 
Theoremnnm0r 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) = ∅)
 
Theoremnnacl 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)
 
Theoremnnmcl 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)
 
Theoremnnacli 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
 
Theoremnnmcli 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
 
Theoremnnacom 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))
 
Theoremnnaass 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)))
 
Theoremnndi 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)))
 
Theoremnnmass 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)))
 
Theoremnnmsucr 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))
 
Theoremnnmcom 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))
 
Theoremnndir 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)))
 
Theoremnnsucelsuc 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 → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
 
Theoremnnsucsssuc 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) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
 
Theoremnntri3or 6382 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 
Theoremnntri2 6383 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremnnsucuniel 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 𝐴𝐵))
 
Theoremnntri1 6385 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremnntri3 6386 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
 
Theoremnntri2or2 6387 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))
 
Theoremnndceq 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)
 
Theoremnndcel 6389 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
 
Theoremnnsseleq 6390 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremnnsssuc 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) → (𝐴𝐵𝐴 ∈ suc 𝐵))
 
Theoremnntr2 6392 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremdcdifsnid 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.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 
Theoremfnsnsplitdc 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.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
 
Theoremfunresdfunsndc 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 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)
 
Theoremnndifsnid 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 ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 
Theoremnnaordi 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) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
 
Theoremnnaord 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) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
 
Theoremnnaordr 6399 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 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) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13235
  Copyright terms: Public domain < Previous  Next >