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Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremodi 8201 Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Theoremomass 8202 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

Theoremoneo 8203 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Theoremomeulem1 8204* Lemma for omeu 8207: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)

Theoremomeulem2 8205 Lemma for omeu 8207: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Theoremomopth2 8206 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))

Theoremomeu 8207* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))

Theoremoen0 8208 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))

Theoremoeordi 8209 Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))

Theoremoeord 8210 Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))

Theoremoecan 8211 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))

Theoremoeword 8212 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Theoremoewordi 8213 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Theoremoewordri 8214 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))

Theoremoeworde 8215 Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))

Theoremoeordsuc 8216 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))

Theoremoelim2 8217* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑥 ∈ (𝐵 ∖ 1o)(𝐴o 𝑥))

Theoremoeoalem 8218 Lemma for oeoa 8219. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴    &   𝐵 ∈ On       (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))

Theoremoeoa 8219 Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))

Theoremoeoelem 8220 Lemma for oeoe 8221. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴       ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))

Theoremoeoe 8221 Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))

Theoremoelimcl 8222 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)
((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))

Theoremoeeulem 8223* Lemma for oeeu 8225. (Contributed by Mario Carneiro, 28-Feb-2013.)
𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}       ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴o 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝑋)))

Theoremoeeui 8224* The division algorithm for ordinal exponentiation. (This version of oeeu 8225 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8207.) (Contributed by Mario Carneiro, 25-May-2015.)
𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}    &   𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)       ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴o 𝐶)) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))

Theoremoeeu 8225* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))

2.4.20  Natural number arithmetic

Theoremnna0 8226 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)

Theoremnnm0 8227 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)

Theoremnnasuc 8228 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 8229 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))

Theoremnnesuc 8230 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Theoremnna0r 8231 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 8159) so that we can avoid ax-rep 5176, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
(𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)

Theoremnnm0r 8232 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 8233 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 8234 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)

Theoremnnecl 8235 Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴o 𝐵) ∈ ω)

Theoremnnacli 8236 ω is closed under addition. Inference form of nnacl 8233. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +o 𝐵) ∈ ω

Theoremnnmcli 8237 ω is closed under multiplication. Inference form of nnmcl 8234. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·o 𝐵) ∈ ω

Theoremnnarcl 8238 Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))

Theoremnnacom 8239 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))

Theoremnnaordi 8240 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 8241 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 8242 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)))

Theoremnnawordi 8243 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)))

Theoremnnaass 8244 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 8245 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 8246 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 8247 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 8248 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))

Theoremnnaword 8249 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 8250 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 8251 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 8252 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))

Theoremnnmordi 8253 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 8254 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 8255 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 8256 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))

Theoremnnmwordi 8257 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Theoremnnmwordri 8258 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))

Theoremnnawordex 8259* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

Theoremnnaordex 8260* 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Theorem1onn 8261 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1o ∈ ω

Theorem2onn 8262 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2o ∈ ω

Theorem3onn 8263 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ ω

Theorem4onn 8264 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ ω

Theorem1one2o 8265 Ordinal one is not ordinal two. Analogous to 1ne2 11842. (Contributed by AV, 17-Oct-2023.)
1o ≠ 2o

Theoremoaabslem 8266 Lemma for oaabs 8267. (Contributed by NM, 9-Dec-2004.)
((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)

Theoremoaabs 8267 Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)

Theoremoaabs2 8268 The absorption law oaabs 8267 is also a property of higher powers of ω. (Contributed by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ (ω ↑o 𝐶) ∧ 𝐵 ∈ On) ∧ (ω ↑o 𝐶) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)

Theoremomabslem 8269 Lemma for omabs 8270. (Contributed by Mario Carneiro, 30-May-2015.)
((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)

Theoremomabs 8270 Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.)
(((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))

Theoremnnm1 8271 Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)

Theoremnnm2 8272 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))

Theoremnn2m 8273 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))

Theoremnnneo 8274 If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Theoremnneob 8275* A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
(𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))

Theoremomsmolem 8276* Lemma for omsmo 8277. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
(𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))

Theoremomsmo 8277* A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
(((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)

Theoremomopthlem1 8278 Lemma for omopthi 8280. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ ω    &   𝐶 ∈ ω       (𝐴𝐶 → ((𝐴 ·o 𝐴) +o (𝐴 ·o 2o)) ∈ (𝐶 ·o 𝐶))

Theoremomopthlem2 8279 Lemma for omopthi 8280. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐶 ∈ ω    &   𝐷 ∈ ω       ((𝐴 +o 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·o 𝐶) +o 𝐷) = (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵))

Theoremomopthi 8280 An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13635. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐶 ∈ ω    &   𝐷 ∈ ω       ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremomopth 8281 An ordered pair theorem for finite integers. Analogous to nn0opthi 13635. (Contributed by Scott Fenton, 1-May-2012.)
(((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ ω)) → ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

2.4.21  Equivalence relations and classes

Syntaxwer 8282 Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴

Syntaxcec 8283 Extend the definition of a class to include equivalence class.
class [𝐴]𝑅

Syntaxcqs 8284 Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)

Definitiondf-er 8285 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 8286 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 8305, ersymb 8299, and ertr 8300. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
(𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))

Theoremdfer2 8286* 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 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))

Definitiondf-ec 8287 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 8286). 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 8288. (Contributed by NM, 23-Jul-1995.)
[𝐴]𝑅 = (𝑅 “ {𝐴})

Theoremdfec2 8288* 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.)
(𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})

Theoremecexg 8289 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
(𝑅𝐵 → [𝐴]𝑅 ∈ V)

Theoremecexr 8290 A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)

Definitiondf-qs 8291* Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
(𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}

Theoremereq1 8292 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 = 𝑆 → (𝑅 Er 𝐴𝑆 Er 𝐴))

Theoremereq2 8293 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝐴 = 𝐵 → (𝑅 Er 𝐴𝑅 Er 𝐵))

Theoremerrel 8294 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → Rel 𝑅)

Theoremerdm 8295 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Theoremercl 8296 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴𝑋)

Theoremersym 8297 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑅𝐴)

Theoremercl2 8298 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑋)

Theoremersymb 8299 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)       (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremertr 8300 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)       (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

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