P. 423 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	naddcnffn 42201	Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
Theorem	naddcnffo 42202	Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)
Theorem	naddcnfcl 42203	Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) ∈ 𝑆)
Theorem	naddcnfcom 42204	Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
Theorem	naddcnfid1 42205	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
Theorem	naddcnfid2 42206	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)
Theorem	naddcnfass 42207	Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
Theorem	onsucunifi 42208*	The successor to the union of any non-empty, finite subset of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥)
Theorem	sucunisn 42209	The successor to the union of any singleton of a set is the successor of the set. (Contributed by RP, 11-Feb-2025.)
⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)
Theorem	onsucunipr 42210	The successor to the union of any pair of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
Theorem	onsucunitp 42211	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 42212*	The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 42220. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)})
Theorem	oaun3lem2 42213*	The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 42220. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Theorem	oaun3lem3 42214*	The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 42220. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ On)
Theorem	oaun3lem4 42215*	The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 42220. (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 42216*	Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
Theorem	oadif1lem 42217*	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 42218*	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 42219*	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 42220*	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 42221*	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 42222*	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 42223*	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 42224*	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 42225*	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 42226*	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 42227*	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 42228*	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 42229*	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 42230	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1o) = suc 𝐴)
Theorem	naddsuc2 42231	Natural addition with successor. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no suc 𝐵) = suc (𝐴 +no 𝐵))
Theorem	naddass1 42232	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 42233	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 42234	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 42235	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 42236	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 42237	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 42238*	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 42239*	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 8559. (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 42240*	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 42241	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 42173 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 42242	The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∩ suc 𝐴) = 𝐴
Theorem	om2 42243	Two ways to double an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 +o 𝐴) = (𝐴 ·o 2o))
Theorem	oaltom 42244	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 42245	Two ways to square an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ·o 𝐴) = (𝐴 ↑o 2o))
Theorem	omltoe 42246	Exponentiation eventually dominates multiplication. (Contributed by RP, 3-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴)))
21.34.3  Surreal Contributions
Theorem	abeqabi 42247	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 42248*	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 42249*	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 42250*	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 42251	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 42252	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 42253	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 42254	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 42255*	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 42256	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 42257*	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 42258	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 42259	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 42260	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 42261*	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 42262*	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 27294. (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 42263*	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 42264*	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 42265*	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 42266*	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 42267	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 42268	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
Theorem	onnobdayg 42269	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 42270	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 42271	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 42272	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No )
Theorem	onnoi 42273	Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 × {2o}) ∈ No
Theorem	0no 42274	Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ ∅ ∈ No
Theorem	1no 42275	Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (1o × {2o}) ∈ No
Theorem	2no 42276	Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (2o × {2o}) ∈ No
Theorem	3no 42277	Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (3o × {2o}) ∈ No
Theorem	4no 42278	Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
⊢ (4o × {2o}) ∈ No
Theorem	fnimafnex 42279	The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)
⊢ 𝐹 Fn 𝐵    ⇒   ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V
21.34.4  Short Studies
Theorem	nlimsuc 42280	A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)
Theorem	nlim1NEW 42281	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 1o
Theorem	nlim2NEW 42282	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 2o
Theorem	nlim3 42283	3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 3o
Theorem	nlim4 42284	4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 4o
Theorem	oa1un 42285	Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8533. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = (𝐴 ∪ {𝐴}))
Theorem	oa1cl 42286	𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) ∈ On)
Theorem	0finon 42287	0 is a finite ordinal. See peano1 7881. (Contributed by RP, 27-Sep-2023.)
⊢ ∅ ∈ (On ∩ Fin)
Theorem	1finon 42288	1 is a finite ordinal. See 1onn 8641. (Contributed by RP, 27-Sep-2023.)
⊢ 1o ∈ (On ∩ Fin)
Theorem	2finon 42289	2 is a finite ordinal. See 1onn 8641. (Contributed by RP, 27-Sep-2023.)
⊢ 2o ∈ (On ∩ Fin)
Theorem	3finon 42290	3 is a finite ordinal. See 1onn 8641. (Contributed by RP, 27-Sep-2023.)
⊢ 3o ∈ (On ∩ Fin)
Theorem	4finon 42291	4 is a finite ordinal. See 1onn 8641. (Contributed by RP, 27-Sep-2023.)
⊢ 4o ∈ (On ∩ Fin)
Theorem	finona1cl 42292	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 42293	The finite ordinals are a set. See also onprc 7767 and fiprc 9047 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.)
⊢ (On ∩ Fin) ∈ V
Theorem	fzunt 42294	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 42295	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 42296	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 42297	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))
21.34.4.1  Additional work on conditional logical operator
Theorem	ifpan123g 42298	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 42299	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 42300	Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))