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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1oex 6401 Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
1o ∈ V
 
Theorem2on 6402 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2o ∈ On
 
Theorem2on0 6403 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2o ≠ ∅
 
Theorem3on 6404 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ On
 
Theorem4on 6405 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ On
 
Theoremdf1o2 6406 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1o = {∅}
 
Theoremdf2o3 6407 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2o = {∅, 1o}
 
Theoremdf2o2 6408 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2o = {∅, {∅}}
 
Theorem1n0 6409 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1o ≠ ∅
 
Theoremxp01disj 6410 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
 
Theoremxp01disjl 6411 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
(({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
 
Theoremordgt0ge1 6412 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
 
Theoremordge1n0im 6413 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
(Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
 
Theoremel1o 6414 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1o𝐴 = ∅)
 
Theoremdif1o 6415 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 6416 Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
 
Theorem0lt1o 6417 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1o
 
Theorem0lt2o 6418 Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
∅ ∈ 2o
 
Theorem1lt2o 6419 Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
1o ∈ 2o
 
Theoremel2oss1o 6420 Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13988. (Contributed by Jim Kingdon, 8-Aug-2022.)
(𝐴 ∈ 2o𝐴 ⊆ 1o)
 
Theoremoafnex 6421 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
(𝑥 ∈ V ↦ suc 𝑥) Fn V
 
Theoremsucinc 6422* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       𝑥 𝑥 ⊆ (𝐹𝑥)
 
Theoremsucinc2 6423* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremfnoa 6424 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 6425 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 +o 𝐵) ∈ V)
 
Theoremomfnex 6426* 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 6427 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
·o Fn (On × On)
 
Theoremomexg 6428 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ·o 𝐵) ∈ V)
 
Theoremfnoei 6429 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 6430 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴o 𝐵) ∈ V)
 
Theoremoav 6431* 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 6432* 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 6433* 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 6434 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 6435 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 6436 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 6437 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 6438 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 6439 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
 
Theoremoav2 6440* 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 6441 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 6442* 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 6443 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 6444 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 6445 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1o +o 1o) = 2o
 
Theoremoawordi 6446 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 6447* A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6446. (Contributed by Jim Kingdon, 15-May-2022.)
((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))       (𝜑 ∨ ¬ 𝜑)
 
Theoremoaword1 6448 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 6449 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 6450 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))
 
2.6.24  Natural number arithmetic
 
Theoremnna0 6451 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
 
Theoremnnm0 6452 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
 
Theoremnnasuc 6453 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 6454 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 6455 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 6456 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 6457 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 6458 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 6459 ω is closed under addition. Inference form of nnacl 6457. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +o 𝐵) ∈ ω
 
Theoremnnmcli 6460 ω is closed under multiplication. Inference form of nnmcl 6458. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·o 𝐵) ∈ ω
 
Theoremnnacom 6461 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 6462 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 6463 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 6464 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 6465 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 6466 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 6467 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 6468 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4490, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4512. (Contributed by Jim Kingdon, 25-Aug-2019.)
(𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
 
Theoremnnsucsssuc 6469 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4491, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4509. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
 
Theoremnntri3or 6470 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 
Theoremnntri2 6471 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremnnsucuniel 6472 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 4492). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4513). (Contributed by Jim Kingdon, 13-Mar-2022.)
(𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))
 
Theoremnntri1 6473 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremnntri3 6474 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
 
Theoremnntri2or2 6475 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))
 
Theoremnndceq 6476 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4600. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
 
Theoremnndcel 6477 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
 
Theoremnnsseleq 6478 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 6479 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 6480 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremdcdifsnid 6481* 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 3724 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 6482* 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 6483* 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 6484 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 3724 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 6485 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 6486 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 6487 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 6488 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 6489 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 6490 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 6491 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
 
Theoremnnawordi 6492 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 6493 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 6494 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 6495 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 6496 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 6497 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1o ∈ ω
 
Theorem2onn 6498 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2o ∈ ω
 
Theorem3onn 6499 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ ω
 
Theorem4onn 6500 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ ω
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