P. 67 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnoa 6601	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 6602	Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +o 𝐵) ∈ V)
Theorem	omfnex 6603*	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 6604	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 6605	Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ·o 𝐵) ∈ V)
Theorem	fnoei 6606	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 6607	Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑o 𝐵) ∈ V)
Theorem	oav 6608*	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 6609*	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 6610*	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 6611	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 6612	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 6613	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 6614	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 6615	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 6616	Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
Theorem	oav2 6617*	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 6618	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 6619*	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 6620	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 6621	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 6622	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
⊢ (1o +o 1o) = 2o
Theorem	oawordi 6623	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 6624*	A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6623. (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 6625	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 6626	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 6627	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
Theorem	nna0 6628	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
Theorem	nnm0 6629	Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
Theorem	nnasuc 6630	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 6631	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 6632	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 6633	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 6634	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 6635	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 6636	ω is closed under addition. Inference form of nnacl 6634. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 +o 𝐵) ∈ ω
Theorem	nnmcli 6637	ω is closed under multiplication. Inference form of nnmcl 6635. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 ·o 𝐵) ∈ ω
Theorem	nnacom 6638	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 6639	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 6640	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 6641	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 6642	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 6643	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 6644	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 6645	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4600, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4622. (Contributed by Jim Kingdon, 25-Aug-2019.)
⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
Theorem	nnsucsssuc 6646	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4601, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4619. (Contributed by Jim Kingdon, 25-Aug-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Theorem	nntri3or 6647	Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	nntri2 6648	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 6649	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 4602). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4623). (Contributed by Jim Kingdon, 13-Mar-2022.)
⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
Theorem	nntri1 6650	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
Theorem	nntri3 6651	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 6652	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	nndceq 6653	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4710. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
Theorem	nndcel 6654	Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 ∈ 𝐵)
Theorem	nnsseleq 6655	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 6656	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 6657	Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	dcdifsnid 6658*	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 3814 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 6659*	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 6660*	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 6661	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 3814 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 6662	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 6663	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 6664	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 6665	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)))
Theorem	nnacan 6666	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))
Theorem	nnaword1 6667	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))
Theorem	nnaword2 6668	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
Theorem	nnawordi 6669	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) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
Theorem	nnmordi 6670	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) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	nnmord 6671	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) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	nnmword 6672	Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
Theorem	nnmcan 6673	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))
Theorem	1onn 6674	One is a natural number. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ ω
Theorem	2onn 6675	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
⊢ 2o ∈ ω
Theorem	3onn 6676	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ ω
Theorem	4onn 6677	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ ω
Theorem	2ssom 6678	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
⊢ 2o ⊆ ω
Theorem	nnm1 6679	Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)
Theorem	nnm2 6680	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))
Theorem	nn2m 6681	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))
Theorem	nnaordex 6682*	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) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Theorem	nnawordex 6683*	Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
Theorem	nnm00 6684	The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
2.6.25  Equivalence relations and classes
Syntax	wer 6685	Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴
Syntax	cec 6686	Extend the definition of a class to include equivalence class.
class [𝐴]𝑅
Syntax	cqs 6687	Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)
Definition	df-er 6688	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 6689 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6708, ersymb 6702, and ertr 6703. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
Theorem	dfer2 6689*	Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
Definition	df-ec 6690	Define the 𝑅-coset of 𝐴. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of 𝐴 modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 6689). In this case, 𝐴 is a representative (member) of the equivalence class [𝐴]𝑅, which contains all sets that are equivalent to 𝐴. Definition of [Enderton] p. 57 uses the notation [𝐴] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 6691. (Contributed by NM, 23-Jul-1995.)
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
Theorem	dfec2 6691*	Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	ecexg 6692	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V)
Theorem	ecexr 6693	An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
Definition	df-qs 6694*	Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
Theorem	ereq1 6695	Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))
Theorem	ereq2 6696	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵))
Theorem	errel 6697	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → Rel 𝑅)
Theorem	erdm 6698	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
Theorem	ercl 6699	Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝑋)
Theorem	ersym 6700	An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐴)