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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoa0 6401 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0 6402 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoei0 6403 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.)
o

Theoremoacl 6404 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.)

Theoremomcl 6405 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.)

Theoremoeicl 6406 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
o

Theoremoav2 6407* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)

Theoremoasuc 6408 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomv2 6409* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)

Theoremonasuc 6410 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremoa1suc 6411 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.)

Theoremo1p1e2 6412 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)

Theoremoawordi 6413 Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)

Theoremoawordriexmid 6414* A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6413. (Contributed by Jim Kingdon, 15-May-2022.)

Theoremoaword1 6415 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.)

Theoremomsuc 6416 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremonmsuc 6417 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

2.6.24  Natural number arithmetic

Theoremnna0 6418 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)

Theoremnnm0 6419 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)

Theoremnnasuc 6420 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnmsuc 6421 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnna0r 6422 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.)

Theoremnnm0r 6423 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacl 6424 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.)

Theoremnnmcl 6425 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.)

Theoremnnacli 6426 is closed under addition. Inference form of nnacl 6424. (Contributed by Scott Fenton, 20-Apr-2012.)

Theoremnnmcli 6427 is closed under multiplication. Inference form of nnmcl 6425. (Contributed by Scott Fenton, 20-Apr-2012.)

Theoremnnacom 6428 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.)

Theoremnnaass 6429 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.)

Theoremnndi 6430 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.)

Theoremnnmass 6431 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.)

Theoremnnmsucr 6432 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnmcom 6433 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.)

Theoremnndir 6434 Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)

Theoremnnsucelsuc 6435 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4466, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4488. (Contributed by Jim Kingdon, 25-Aug-2019.)

Theoremnnsucsssuc 6436 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4467, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4485. (Contributed by Jim Kingdon, 25-Aug-2019.)

Theoremnntri3or 6437 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)

Theoremnntri2 6438 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)

Theoremnnsucuniel 6439 Given an element of the union of a natural number , is an element of itself. The reverse direction holds for all ordinals (sucunielr 4468). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4489). (Contributed by Jim Kingdon, 13-Mar-2022.)

Theoremnntri1 6440 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)

Theoremnntri3 6441 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)

Theoremnntri2or2 6442 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)

Theoremnndceq 6443 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where is zero, see nndceq0 4576. (Contributed by Jim Kingdon, 31-Aug-2019.)
DECID

Theoremnndcel 6444 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
DECID

Theoremnnsseleq 6445 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)

Theoremnnsssuc 6446 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.)

Theoremnntr2 6447 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)

Theoremdcdifsnid 6448* 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 3702 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.)
DECID

Theoremfnsnsplitdc 6449* 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.)
DECID

Theoremfunresdfunsndc 6450* 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.)
DECID

Theoremnndifsnid 6451 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 3702 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)

Theoremnnaordi 6452 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.)

Theoremnnaord 6453 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.)

Theoremnnaordr 6454 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)

Theoremnnaword 6455 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacan 6456 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword1 6457 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword2 6458 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)

Theoremnnawordi 6459 Adding to both sides of an inequality in (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnmordi 6460 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.)

Theoremnnmord 6461 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.)

Theoremnnmword 6462 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmcan 6463 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorem1onn 6464 One is a natural number. (Contributed by NM, 29-Oct-1995.)

Theorem2onn 6465 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)

Theorem3onn 6466 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4onn 6467 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremnnm1 6468 Multiply an element of by . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnm2 6469 Multiply an element of by . (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnn2m 6470 Multiply an element of by . (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnaordex 6471* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnawordex 6472* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnm00 6473 The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)

2.6.25  Equivalence relations and classes

Syntaxwer 6474 Extend the definition of a wff to include the equivalence predicate.

Syntaxcec 6475 Extend the definition of a class to include equivalence class.

Syntaxcqs 6476 Extend the definition of a class to include quotient set.

Definitiondf-er 6477 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6478 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6497, ersymb 6491, and ertr 6492. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)

Theoremdfer2 6478* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)

Definitiondf-ec 6479 Define the -coset of . Exercise 35 of [Enderton] p. 61. This is called the equivalence class of modulo when is an equivalence relation (i.e. when ; see dfer2 6478). In this case, is a representative (member) of the equivalence class , which contains all sets that are equivalent to . Definition of [Enderton] p. 57 uses the notation (subscript) , although we simply follow the brackets by since we don't have subscripted expressions. For an alternate definition, see dfec2 6480. (Contributed by NM, 23-Jul-1995.)

Theoremdfec2 6480* Alternate definition of -coset of . Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theoremecexg 6481 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)

Theoremecexr 6482 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Definitiondf-qs 6483* Define quotient set. is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)

Theoremereq1 6484 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremereq2 6485 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrel 6486 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerdm 6487 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremercl 6488 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersym 6489 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercl2 6490 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersymb 6491 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertr 6492 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertrd 6493 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr2d 6494 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr3d 6495 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr4d 6496 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremerref 6497 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercnv 6498 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrn 6499 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerssxp 6500 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)

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