P. 439 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsucunitp 43801	The successor to the union of any triple of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
Theorem	oaun3lem1 43802*	The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 43810. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)})
Theorem	oaun3lem2 43803*	The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 43810. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Theorem	oaun3lem3 43804*	The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 43810. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ On)
Theorem	oaun3lem4 43805*	The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 43810. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
Theorem	rp-abid 43806*	Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
Theorem	oadif1lem 43807*	Express the set difference of a continuous sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊕ 𝐵) ∈ On)    &   ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 ⊕ 𝑏) ∈ On)    &   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦)    &   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵)))    &   ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 ⊕ 𝑏))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ⊕ 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)})
Theorem	oadif1 43808*	Express the set difference of an ordinal sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)})
Theorem	oaun2 43809*	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}})
Theorem	oaun3 43810*	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
Theorem	naddov4 43811*	Alternate expression for natural addition. (Contributed by RP, 19-Dec-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))
Theorem	nadd2rabtr 43812*	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 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Theorem	nadd2rabord 43813*	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 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Theorem	nadd2rabex 43814*	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 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ V)
Theorem	nadd2rabon 43815*	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 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ On)
Theorem	nadd1rabtr 43816*	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 43817*	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 43818*	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 43819*	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 43820	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1o) = suc 𝐴)
Theorem	naddass1 43821	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 43822	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 43823	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 43824	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 43825	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 43826	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 43827*	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 43828*	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 8492. (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 43829*	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 43830	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 43763 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 43831	The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∩ suc 𝐴) = 𝐴
Theorem	oaltom 43832	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 43833	Two ways to square an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ·o 𝐴) = (𝐴 ↑o 2o))
Theorem	omltoe 43834	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 43835	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 43836*	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 43837*	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 43838*	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 43839	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 43840	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 43841	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 43842	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 43843*	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 43844	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 43845*	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 43846	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 43847	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 43848	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 43849*	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 43850*	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 brslts 27754. (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 43851*	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 43852*	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 43853*	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 43854*	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 43855	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	onnoxpg 43856	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
Theorem	onnobdayg 43857	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 43858	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 43859	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	onnoxp 43860	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No )
Theorem	onnoxpi 43861	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 × {2o}) ∈ No
Theorem	0fno 43862	Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ∅ ∈ No
Theorem	1fno 43863	Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (1o × {2o}) ∈ No
Theorem	2fno 43864	Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (2o × {2o}) ∈ No
Theorem	3fno 43865	Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (3o × {2o}) ∈ No
Theorem	4fno 43866	Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (4o × {2o}) ∈ No
Theorem	fnimafnex 43867	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 43868	A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)
Theorem	nlim1NEW 43869	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 1o
Theorem	nlim2NEW 43870	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 2o
Theorem	nlim3 43871	3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 3o
Theorem	nlim4 43872	4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 4o
Theorem	oa1un 43873	Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8466. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = (𝐴 ∪ {𝐴}))
Theorem	oa1cl 43874	𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) ∈ On)
Theorem	0finon 43875	0 is a finite ordinal. See peano1 7840. (Contributed by RP, 27-Sep-2023.)
⊢ ∅ ∈ (On ∩ Fin)
Theorem	1finon 43876	1 is a finite ordinal. See 1onn 8576. (Contributed by RP, 27-Sep-2023.)
⊢ 1o ∈ (On ∩ Fin)
Theorem	2finon 43877	2 is a finite ordinal. See 1onn 8576. (Contributed by RP, 27-Sep-2023.)
⊢ 2o ∈ (On ∩ Fin)
Theorem	3finon 43878	3 is a finite ordinal. See 1onn 8576. (Contributed by RP, 27-Sep-2023.)
⊢ 3o ∈ (On ∩ Fin)
Theorem	4finon 43879	4 is a finite ordinal. See 1onn 8576. (Contributed by RP, 27-Sep-2023.)
⊢ 4o ∈ (On ∩ Fin)
Theorem	finona1cl 43880	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 43881	The finite ordinals are a set. See also onprc 7732 and fiprc 8991 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.)
⊢ (On ∩ Fin) ∈ V
Theorem	fzunt 43882	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 43883	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 43884	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 43885	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 43886	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 43887	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 43888	Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))
Theorem	ifporcor 43889	Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
⊢ (if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))
Theorem	ifpdfan2 43890	Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
Theorem	ifpancor 43891	Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
Theorem	ifpdfor 43892	Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓))
Theorem	ifpdfan 43893	Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥))
Theorem	ifpbi2 43894	Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
Theorem	ifpbi3 43895	Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))
Theorem	ifpim1 43896	Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
Theorem	ifpnot 43897	Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤))
Theorem	ifpid2 43898	Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥))
Theorem	ifpim2 43899	Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
Theorem	ifpbi23 43900	Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))