P. 435 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43401-43500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nadd1rabtr 43401*	The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})
Theorem	nadd1rabord 43402*	The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})
Theorem	nadd1rabex 43403*	The class of ordinals which have a natural sum less than some ordinal is a set. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ V)
Theorem	nadd1rabon 43404*	The set of ordinals which have a natural sum less than some ordinal is an ordinal number. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ On)
Theorem	nadd1suc 43405	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1o) = suc 𝐴)
Theorem	naddass1 43406	Natural addition of ordinal numbers is associative when the third element is 1. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) +no 1o) = (𝐴 +no (𝐵 +no 1o)))
Theorem	naddgeoa 43407	Natural addition results in a value greater than or equal than that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))
Theorem	naddonnn 43408	Natural addition with a natural number on the right results in a value equal to that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))
Theorem	naddwordnexlem0 43409	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, (ω ·o suc 𝐶) lies between 𝐴 and 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
Theorem	naddwordnexlem1 43410	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is equal to or larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	naddwordnexlem2 43411	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	naddwordnexlem3 43412*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, every natural sum of 𝐴 with a natural number is less that 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐴 +no 𝑥) ∈ 𝐵)
Theorem	oawordex3 43413*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, some ordinal sum of 𝐴 is equal to 𝐵. This is a specialization of oawordex 8595. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Theorem	naddwordnexlem4 43414*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, there exists a product with omega such that the ordinal sum with 𝐴 is less than or equal to 𝐵 while the natural sum is larger than 𝐵. (Contributed by RP, 15-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    &   ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1o)((𝐶 +o 𝑥) = 𝐷 ∧ (𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o 𝑥))))
Theorem	ordsssucim 43415	If an ordinal is less than or equal to the successor of another, then the first is either less than or equal to the second or the first is equal to the successor of the second. Theorem 1 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 See also ordsssucb 43348 for a biimplication when 𝐴 is a set. (Contributed by RP, 3-Jan-2025.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	insucid 43416	The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∩ suc 𝐴) = 𝐴
Theorem	om2 43417	Two ways to double an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 +o 𝐴) = (𝐴 ·o 2o))
Theorem	oaltom 43418	Multiplication eventually dominates addition. (Contributed by RP, 3-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))
Theorem	oe2 43419	Two ways to square an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ·o 𝐴) = (𝐴 ↑o 2o))
Theorem	omltoe 43420	Exponentiation eventually dominates multiplication. (Contributed by RP, 3-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴)))
21.36.3  Surreal Contributions
Theorem	abeqabi 43421	Generalized condition for a class abstraction to be equal to some class. (Contributed by RP, 2-Sep-2024.)
⊢ 𝐴 = {𝑥 ∣ 𝜓}    ⇒   ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))
Theorem	abpr 43422*	Condition for a class abstraction to be a pair. (Contributed by RP, 25-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = {𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))
Theorem	abtp 43423*	Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))
Theorem	ralopabb 43424*	Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒))
Theorem	fpwfvss 43425	Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.)
⊢ 𝐹:𝐶⟶𝒫 𝐵    ⇒   ⊢ (𝐹‘𝐴) ⊆ 𝐵
Theorem	sdomne0 43426	A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.)
⊢ (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅)
Theorem	sdomne0d 43427	A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024.)
⊢ (𝜑 → 𝐵 ≺ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	safesnsupfiss 43428	If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵)
Theorem	safesnsupfiub 43429*	If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦 ∈ 𝐶 𝑥𝑅𝑦)
Theorem	safesnsupfidom1o 43430	If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
Theorem	safesnsupfilb 43431*	If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 3-Sep-2024.)
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦)
Theorem	isoeq145d 43432	Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
Theorem	resisoeq45d 43433	Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))
Theorem	negslem1 43434	An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.)
⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ◡𝑅(𝐵, 𝐵)))
Theorem	nvocnvb 43435*	Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))
Theorem	rp-brsslt 43436*	Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27830. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.)
⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}    ⇒   ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)))
Theorem	nla0002 43437*	Extending a linear order to subsets, the empty set is less than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → ∅ < 𝐴)
Theorem	nla0003 43438*	Extending a linear order to subsets, the empty set is greater than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝐴 < ∅)
Theorem	nla0001 43439*	Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}    ⇒   ⊢ (𝜑 → ∅ < ∅)
Theorem	faosnf0.11b 43440*	𝐵 is called a non-limit ordinal if it is not a limit ordinal. (Contributed by RP, 27-Sep-2023.) Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 565-73. http://www.jstor.org/stable/44237243.
⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Theorem	dfno2 43441	A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
⊢ No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
Theorem	onnog 43442	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
Theorem	onnobdayg 43443	Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)
Theorem	bdaybndex 43444	Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.)
⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
Theorem	bdaybndbday 43445	Bounds formed from the birthday have the same birthday. (Contributed by RP, 30-Sep-2023.)
⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))
Theorem	onno 43446	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No )
Theorem	onnoi 43447	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 × {2o}) ∈ No
Theorem	0no 43448	Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ∅ ∈ No
Theorem	1no 43449	Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (1o × {2o}) ∈ No
Theorem	2no 43450	Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (2o × {2o}) ∈ No
Theorem	3no 43451	Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (3o × {2o}) ∈ No
Theorem	4no 43452	Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (4o × {2o}) ∈ No
Theorem	fnimafnex 43453	The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)
⊢ 𝐹 Fn 𝐵    ⇒   ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V
21.36.4  Short Studies
Theorem	nlimsuc 43454	A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)
Theorem	nlim1NEW 43455	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 1o
Theorem	nlim2NEW 43456	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 2o
Theorem	nlim3 43457	3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 3o
Theorem	nlim4 43458	4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 4o
Theorem	oa1un 43459	Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8569. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = (𝐴 ∪ {𝐴}))
Theorem	oa1cl 43460	𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) ∈ On)
Theorem	0finon 43461	0 is a finite ordinal. See peano1 7910. (Contributed by RP, 27-Sep-2023.)
⊢ ∅ ∈ (On ∩ Fin)
Theorem	1finon 43462	1 is a finite ordinal. See 1onn 8678. (Contributed by RP, 27-Sep-2023.)
⊢ 1o ∈ (On ∩ Fin)
Theorem	2finon 43463	2 is a finite ordinal. See 1onn 8678. (Contributed by RP, 27-Sep-2023.)
⊢ 2o ∈ (On ∩ Fin)
Theorem	3finon 43464	3 is a finite ordinal. See 1onn 8678. (Contributed by RP, 27-Sep-2023.)
⊢ 3o ∈ (On ∩ Fin)
Theorem	4finon 43465	4 is a finite ordinal. See 1onn 8678. (Contributed by RP, 27-Sep-2023.)
⊢ 4o ∈ (On ∩ Fin)
Theorem	finona1cl 43466	The finite ordinals are closed under the add one operation. (Contributed by RP, 27-Sep-2023.)
⊢ (𝑁 ∈ (On ∩ Fin) → (𝑁 +o 1o) ∈ (On ∩ Fin))
Theorem	finonex 43467	The finite ordinals are a set. See also onprc 7798 and fiprc 9085 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.)
⊢ (On ∩ Fin) ∈ V
Theorem	fzunt 43468	Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
Theorem	fzuntd 43469	Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
Theorem	fzunt1d 43470	Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    &   ⊢ (𝜑 → 𝑀 ≤ 𝐿)    &   ⊢ (𝜑 → 𝐿 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
Theorem	fzuntgd 43471	Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    &   ⊢ (𝜑 → 𝑀 ≤ (𝐿 + 1))    &   ⊢ (𝜑 → 𝐿 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
21.36.4.1  Additional work on conditional logical operator
Theorem	ifpan123g 43472	Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂))))
Theorem	ifpan23 43473	Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏)))
Theorem	ifpdfor2 43474	Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))
Theorem	ifporcor 43475	Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
⊢ (if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))
Theorem	ifpdfan2 43476	Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
Theorem	ifpancor 43477	Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
Theorem	ifpdfor 43478	Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓))
Theorem	ifpdfan 43479	Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥))
Theorem	ifpbi2 43480	Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
Theorem	ifpbi3 43481	Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))
Theorem	ifpim1 43482	Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
Theorem	ifpnot 43483	Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤))
Theorem	ifpid2 43484	Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥))
Theorem	ifpim2 43485	Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
Theorem	ifpbi23 43486	Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
Theorem	ifpbiidcor 43487	Restatement of biid 261. (Contributed by RP, 25-Apr-2020.)
⊢ if-(𝜑, 𝜑, ¬ 𝜑)
Theorem	ifpbicor 43488	Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
Theorem	ifpxorcor 43489	Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))
Theorem	ifpbi1 43490	Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))
Theorem	ifpnot23 43491	Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
Theorem	ifpnotnotb 43492	Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))
Theorem	ifpnorcor 43493	Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑))
Theorem	ifpnancor 43494	Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))
Theorem	ifpnot23b 43495	Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))
Theorem	ifpbiidcor2 43496	Restatement of biid 261. (Contributed by RP, 25-Apr-2020.)
⊢ ¬ if-(𝜑, ¬ 𝜑, 𝜑)
Theorem	ifpnot23c 43497	Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))
Theorem	ifpnot23d 43498	Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))
Theorem	ifpdfnan 43499	Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ⊼ 𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤))
Theorem	ifpdfxor 43500	Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ⊻ 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓))