P. 85 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rdgsucmpt2 8401*	This version of rdgsucmpt 8402 avoids the not-free hypothesis of rdgsucmptf 8399 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)    &   ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	rdgsucmpt 8402*	The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)
⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	rdglim2 8403*	The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
Theorem	rdglim2a 8404*	The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑥))
Theorem	rdg0n 8405	If 𝐴 is a proper class, then the recursive function generator at ∅ is the empty set. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ (¬ 𝐴 ∈ V → (rec(𝐹, 𝐴)‘∅) = ∅)
2.4.22  Finite recursion
Theorem	frfnom 8406	The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
Theorem	fr0g 8407	The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
Theorem	frsuc 8408	The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)))
Theorem	frsucmpt 8409	The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	frsucmptn 8410	The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 8409 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Theorem	frsucmpt2 8411*	The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)    &   ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	tz7.48lem 8412*	A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴))
Theorem	tz7.48-2 8413*	Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)
Theorem	tz7.48-1 8414*	Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)
Theorem	tz7.48-3 8415*	Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)
Theorem	tz7.49 8416*	Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
⊢ 𝐹 Fn On    &   ⊢ (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
Theorem	tz7.49c 8417*	Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
⊢ 𝐹 Fn On    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
Syntax	cseqom 8418	Extend class notation to include index-aware recursive definitions.
class seqω(𝐹, 𝐼)
Definition	df-seqom 8419*	Index-aware recursive definitions over ω. A mashup of df-rdg 8381 and df-seq 13974, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
Theorem	seqomlem0 8420*	Lemma for seqω. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
Theorem	seqomlem1 8421*	Lemma for seqω. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
Theorem	seqomlem2 8422*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝑄 “ ω) Fn ω
Theorem	seqomlem3 8423*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
Theorem	seqomlem4 8424*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Theorem	seqomeq12 8425	Equality theorem for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))
Theorem	fnseqom 8426	An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ 𝐺 Fn ω
Theorem	seqom0g 8427	Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by AV, 17-Sep-2021.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼)
Theorem	seqomsuc 8428	Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = (𝐴𝐹(𝐺‘𝐴)))
Theorem	omsucelsucb 8429	Membership is inherited by successors for natural numbers. (Contributed by AV, 15-Sep-2023.)
⊢ (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω)
2.4.23  Ordinal arithmetic
Syntax	c1o 8430	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 8431	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	c3o 8432	Extend the definition of a class to include the ordinal number 3.
class 3o
Syntax	c4o 8433	Extend the definition of a class to include the ordinal number 4.
class 4o
Syntax	coa 8434	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 8435	Extend the definition of a class to include the ordinal multiplication operation.
class ·o
Syntax	coe 8436	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 8437	Define the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.)
⊢ 1o = suc ∅
Definition	df-2o 8438	Define the ordinal number 2. Lemma 3.17 of [Schloeder] p. 10. (Contributed by NM, 18-Feb-2004.)
⊢ 2o = suc 1o
Definition	df-3o 8439	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 3o = suc 2o
Definition	df-4o 8440	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 4o = suc 3o
Definition	df-oadd 8441*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
Definition	df-omul 8442*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Definition	df-oexp 8443*	Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
Theorem	df1o2 8444	Expanded value of the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 4-Nov-2002.)
⊢ 1o = {∅}
Theorem	df2o3 8445	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 2o = {∅, 1o}
Theorem	df2o2 8446	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
⊢ 2o = {∅, {∅}}
Theorem	1oex 8447	Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178, ax-un 7714. (Revised by Zhi Wang, 19-Sep-2024.)
⊢ 1o ∈ V
Theorem	2oex 8448	2o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178, ax-un 7714. (Proof shortened by Zhi Wang, 19-Sep-2024.)
⊢ 2o ∈ V
Theorem	1on 8449	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7714. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ 1o ∈ On
Theorem	2on 8450	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Avoid ax-un 7714. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ 2o ∈ On
Theorem	2on0 8451	Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ 2o ≠ ∅
Theorem	ord3 8452	Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
⊢ Ord 3o
Theorem	3on 8453	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ On
Theorem	4on 8454	Ordinal 4 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ On
Theorem	1n0 8455	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
⊢ 1o ≠ ∅
Theorem	nlim1 8456	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ¬ Lim 1o
Theorem	nlim2 8457	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ¬ Lim 2o
Theorem	xp01disj 8458	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Theorem	xp01disjl 8459	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Theorem	ordgt0ge1 8460	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
Theorem	ordge1n0 8461	An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	el1o 8462	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
Theorem	ord1eln01 8463	An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))
Theorem	ord2eln012 8464	An ordinal that is not 0, 1, or 2 contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Ord 𝐴 → (2o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o)))
Theorem	1ellim 8465	A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Lim 𝐴 → 1o ∈ 𝐴)
Theorem	2ellim 8466	A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Lim 𝐴 → 2o ∈ 𝐴)
Theorem	dif1o 8467	Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
Theorem	ondif1 8468	Two ways to say that 𝐴 is a nonzero ordinal number. Lemma 1.10 of [Schloeder] p. 2. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
Theorem	ondif2 8469	Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
Theorem	2oconcl 8470	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)
Theorem	0lt1o 8471	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ ∅ ∈ 1o
Theorem	dif20el 8472	An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
Theorem	0we1 8473	The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ ∅ We 1o
Theorem	brwitnlem 8474	Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o))    &   ⊢ 𝑂 Fn 𝑋    ⇒   ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)
Theorem	fnoa 8475	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ +o Fn (On × On)
Theorem	fnom 8476	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ·o Fn (On × On)
Theorem	fnoe 8477	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
⊢ ↑o Fn (On × On)
Theorem	oav 8478*	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 8479*	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	oe0lem 8480	A helper lemma for oe0 8489 and others. (Contributed by NM, 6-Jan-2005.)
⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝜓)    &   ⊢ (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Theorem	oev 8481*	Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Theorem	oevn0 8482*	Value of ordinal exponentiation at a nonzero base. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
Theorem	oa0 8483	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
Theorem	om0 8484	Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. See om0x 8486 for a way to remove the antecedent 𝐴 ∈ On. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
Theorem	oe0m 8485	Value of zero raised to an ordinal. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
Theorem	om0x 8486	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8484, this version works whether or not 𝐴 is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.) (New usage is discouraged.)
⊢ (𝐴 ·o ∅) = ∅
Theorem	oe0m0 8487	Ordinal exponentiation with zero base and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
⊢ (∅ ↑o ∅) = 1o
Theorem	oe0m1 8488	Ordinal exponentiation with zero base and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 5-Jan-2005.)
⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅))
Theorem	oe0 8489	Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
Theorem	oev2 8490*	Alternate value of ordinal exponentiation. Compare oev 8481. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ ∩ 𝐴) ∪ ∩ 𝐵)))
Theorem	oasuc 8491	Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. Definition 2.3 of [Schloeder] p. 4. (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	oesuclem 8492*	Lemma for oesuc 8494. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ Lim 𝑋    &   ⊢ (𝐵 ∈ 𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	omsuc 8493	Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. (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	oesuc 8494	Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	onasuc 8495	Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 8491 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Theorem	onmsuc 8496	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))
Theorem	onesuc 8497	Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
Theorem	oa1suc 8498	Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. Remark 2.4 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
Theorem	oalim 8499*	Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
Theorem	omlim 8500*	Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))