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Theorem List for Metamath Proof Explorer - 42901-43000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnpw2blenfzo2 42901 A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
(𝑁 ∈ ℕ → (𝑁 = (2↑((#b𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b𝑁) − 1)) + 1)..^(2↑(#b𝑁)))))
 
Theoremnnpw2pmod 42902 Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ → 𝑁 = ((2↑((#b𝑁) − 1)) + (𝑁 mod (2↑((#b𝑁) − 1)))))
 
Theoremblen1 42903 The binary length of 1. (Contributed by AV, 21-May-2020.)
(#b‘1) = 1
 
Theoremblen2 42904 The binary length of 2. (Contributed by AV, 21-May-2020.)
(#b‘2) = 2
 
Theoremnnpw2p 42905* Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ → ∃𝑖 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))
 
Theoremnnpw2pb 42906* A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
(𝑁 ∈ ℕ ↔ ∃𝑖 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))
 
Theoremblen1b 42907 The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ0 → ((#b𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1)))
 
Theoremblennnt2 42908 The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.)
(𝑁 ∈ ℕ → (#b‘(2 · 𝑁)) = ((#b𝑁) + 1))
 
Theoremnnolog2flm1 42909 The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ) → (⌊‘(2 logb 𝑁)) = (⌊‘(2 logb (𝑁 − 1))))
 
Theoremblennn0em1 42910 The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.)
((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘(𝑁 / 2)) = ((#b𝑁) − 1))
 
Theoremblennngt2o2 42911 The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (#b𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1))
 
Theoremblengt1fldiv2p1 42912 The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.)
(𝑁 ∈ (ℤ‘2) → (#b𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))
 
Theoremblennn0e2 42913 The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.)
((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b𝑁) = ((#b‘(𝑁 / 2)) + 1))
 
20.36.17.10  Digits

Generalisation of df-bits 15348. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 42933: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)).

 
Syntaxcdig 42914 Extend class notation with the class of the digit extraction operation.
class digit
 
Definitiondf-dig 42915* Definition of an operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝑏. 𝑘 = − 1 corresponds to the first digit of the fractional part (for 𝑏 = 10 the first digit after the decimal point), 𝑘 = 0 corresponds to the last digit of the integer part (for 𝑏 = 10 the first digit before the decimal point). See also digit1 13201. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.)
digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
 
Theoremdigfval 42916* Operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
(𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵)))
 
Theoremdigval 42917 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵))
 
Theoremdigvalnn0 42918 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0)
 
Theoremnn0digval 42919 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵𝐾))) mod 𝐵))
 
Theoremdignn0fr 42920 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)𝑁) = 0)
 
Theoremdignn0ldlem 42921 Lemma for dignnld 42922. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))
 
Theoremdignnld 42922 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0)
 
Theoremdig2nn0ld 42923 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘(#b𝑁))) → (𝐾(digit‘2)𝑁) = 0)
 
Theoremdig2nn1st 42924 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
(𝑁 ∈ ℕ → (((#b𝑁) − 1)(digit‘2)𝑁) = 1)
 
Theoremdig0 42925 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
 
Theoremdigexp 42926 The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵𝑁)) = if(𝐾 = 𝑁, 1, 0))
 
Theoremdig1 42927 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0))
 
Theorem0dig1 42928 The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
(𝐵 ∈ (ℤ‘2) → (0(digit‘𝐵)1) = 1)
 
Theorem0dig2pr01 42929 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
(𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)
 
Theoremdig2nn0 42930 A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})
 
Theorem0dig2nn0e 42931 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0)
 
Theorem0dig2nn0o 42932 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 1)
 
Theoremdig2bits 42933 The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))
 
20.36.17.11  Nonnegative integer as sum of its shifted digits
 
Theoremdignn0flhalflem1 42934 Lemma 1 for dignn0flhalf 42937. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
 
Theoremdignn0flhalflem2 42935 Lemma 2 for dignn0flhalf 42937. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
 
Theoremdignn0ehalf 42936 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
(((𝐴 / 2) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))
 
Theoremdignn0flhalf 42937 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2))))
 
Theoremnn0sumshdiglemA 42938* Lemma for nn0sumshdig 42942 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglemB 42939* Lemma for nn0sumshdig 42942 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem1 42940* Lemma 1 for nn0sumshdig 42942 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem2 42941* Lemma 2 for nn0sumshdig 42942. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
 
Theoremnn0sumshdig 42942* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))
 
20.36.17.12  Algorithms for the multiplication of nonnegative integers
 
Theoremnn0mulfsum 42943* Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)
 
Theoremnn0mullong 42944* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15419. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
 
20.37  Mathbox for Emmett Weisz
 
20.37.1  Miscellaneous Theorems

Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.

 
Theoremnfintd 42945 Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
Theoremnfiund 42946 Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)
 
Theoremiunord 42947* The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7132, but does not use it directly, since ssorduni 7132 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
(∀𝑥𝐴 Ord 𝐵 → Ord 𝑥𝐴 𝐵)
 
Theoremiunordi 42948* The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.)
Ord 𝐵       Ord 𝑥𝐴 𝐵
 
Theoremrspcdf 42949* Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremspd 42950 Specialization deduction, using implicit substitution. Based on the proof of spimed 2417. (Contributed by Emmett Weisz, 17-Jan-2020.)
(𝜒 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (∀𝑥𝜑𝜓))
 
Theoremspcdvw 42951* A version of spcdv 3442 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
(𝜑𝐴𝐵)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremtfis2d 42952* Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
(𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))    &   (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜓)))       (𝜑 → (𝑥 ∈ On → 𝜓))
 
Theorembnd2d 42953* Deduction form of bnd2 8920. (Contributed by Emmett Weisz, 19-Jan-2021.)
(𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜓))
 
Theoremdffun3f 42954* Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴       (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 
20.37.2  Set Recursion
 
20.37.2.1  Basic Properties of Set Recursion

Symbols in this section:

All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 42956. The class 𝑌 is explained in the comment of setrec1lem1 42959. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.

 
Syntaxcsetrecs 42955 Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
 
Definitiondf-setrecs 42956* Define a class setrecs(𝐹) by transfinite recursion, where (𝐹𝑥) is the set of new elements to add to the class given the set 𝑥 of elements in the class so far. We do not need a base case, because we can start with the empty set, which is vacuously a subset of setrecs(𝐹). The goal of this definition is to construct a class fulfilling theorems setrec1 42963 and setrec2v 42968, which give a more intuitive idea of the meaning of setrecs. Unlike wrecs, setrecs is well-defined for any 𝐹 and meaningful for any function 𝐹.

For example, see theorem onsetrec 42979 for how the class On is defined recursively using the successor function.

The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements.

Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2𝑜} would fail the condition since 2𝑜 is not an element of the successor of or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1𝑜} fails the condition, since the only subset of previously generated elements is , and suc ∅ does not have 1𝑜 as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with and repeatedly applying the successor function.

A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ran recs((𝑔 ∈ V ↦ (𝐹 ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 42963 and setrec2v 42968 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Conway noted in the Appendix to Part Zero of ONAG, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant." Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis.

Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct.

Formalizing surreal numbers within metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the metamath framework.

The difficulty in writing a definition in metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 42963 and setrec2v 42968), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 7659, this is not actually a requirement of the metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution.

We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time.

Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction.

If setrecs(𝐹) = 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹.

One necessary condition is that each element of a given 𝑥𝐴 must be generated by 𝐹 when applied to a previous element 𝑦𝐴. In symbols, 𝑥𝐴𝑦𝐴(𝑦𝑥𝑥 ⊆ (𝐹𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular.

A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴.

The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥¬ (𝐹𝑧) ⊆ 𝑦)).

On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝑦𝑧)}

In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. {𝑦 ∣ ∀𝑧 (𝑦𝑧 → ∃𝑤(𝑤𝑦 ∧ (𝐹𝑤) ∈ 𝑧 ∧ ¬ 𝑤𝑧))}

These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurence of 𝑧 with 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1.

There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤𝑧, so the definition requires the additional constraint (𝑎𝑏 → (𝐹 a ) C_ ( F 𝑏)) in order to work. This issue can also be avoided by replacing 𝑧 with 𝑧. The induction version of the result is used in the final definition.

Version 2: (18-Aug-2020) Induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(𝑦𝑧 ≠ ∅ → 𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ (𝐹𝑤) ∩ 𝑧 ≠ ∅))}

In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹𝑤) for 𝑤 ⊆ (𝑦𝑧) even if those elements of (𝐹𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹𝑤) in the definition by ((𝐹𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦𝑧 would be false. We could then simply add all elements of (𝐹𝑤) outside of 𝑦 for any 𝑤𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the metamath definition. The other direction is easier. If a certain 𝑧 fails the metamath definition, then all (𝐹𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦𝑧), and in particular ((𝐹𝑤) ∩ 𝑦) ⊆ 𝑧.

The foundedness version is starting to look more like ax-reg 8653! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2.

Version 3 - Foundedness (5-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → ∃𝑣𝑧𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ 𝑣 ∈ (𝐹𝑤)))}

Now, if we replace (𝐹𝑤) by ((𝐹𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣𝑦 since 𝑣𝑧 and 𝑧𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find

To make our definition look exactly like df-fr 5208, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧.

Version 4 - Foundedness (6-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → 𝑣𝑧𝑤𝑢𝑧(𝑤𝑦 ∧ ¬ 𝑢𝑤𝑣 ∈ (𝐹𝑤))

This is so close to df-fr 5208; the only change needed is to switch 𝑤 with 𝑢𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 8653. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧.

Finally, here's a different dead end I followed:

To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦𝑧 rather than only 𝑦𝑧. This will help improve the foundedness version.

Version 1.1 (28-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤 𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(∃𝑎(𝑎𝑦𝑎𝑧) → ∃𝑤(𝑤𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. )

Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of 𝑧. I think this has the same problem.

We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition:

Version 1.2 (31-Aug-2020) Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

Now this looks more like df-fr 5208! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5208.

This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧.

Version 1.3 (31-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤 𝑧 → (𝑤𝑧 ∧ (𝐹𝑤) ∈ 𝑧))) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 𝑤 𝑧 = ∅ ∧ (𝑤𝑧 ∨ (𝐹𝑤) ∈ 𝑧)))}

𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work.

Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹𝑤) corresponds to the variable 𝑦 in df-fr 5208. Thus, in our case, df-fr 5208 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧𝑣𝑅(𝐹𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → 𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧(𝐹𝑤) ∈ (𝐹𝑣)))

This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F.

Consider the foundedness form of Version 1. We want to show ¬ 𝑤𝑧 ↔ ∀𝑣𝑧¬ 𝑣𝑅(𝐹𝑤) so we can replace one with the other. Negate both sides: 𝑤𝑧 ↔ ∃𝑣𝑧𝑣𝑅(𝐹𝑤)

If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.)

setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
 
Theoremsetrecseq 42957 Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
(𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
 
Theoremnfsetrecs 42958 Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
𝑥𝐹       𝑥setrecs(𝐹)
 
Theoremsetrec1lem1 42959* Lemma for setrec1 42963. This is a utility theorem showing the equivalence of the statement 𝑋𝑌 and its expanded form. The proof uses elabg 3502 and equivalence theorems.

Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)

𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
 
Theoremsetrec1lem2 42960* Lemma for setrec1 42963. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3741, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)    &   (𝜑𝑋𝑌)       (𝜑 𝑋𝑌)
 
Theoremsetrec1lem3 42961* Lemma for setrec1 42963. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 42960. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴𝑌. I don't know if proving this fact directly using setrec1lem1 42959 would be any easier than the current proof using setrec1lem2 42960, and it would only slightly simplify the proof of setrec1 42963. Other than the use of bnd2d 42953, this is a purely technical theorem for rearranging notation from that of setrec1lem2 42960 to that of setrec1 42963. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌))       (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
 
Theoremsetrec1lem4 42962* Lemma for setrec1 42963. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 42963, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis 𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1910, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 42960 could similarly be made easier to read by adding the hypothesis 𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.)

𝑧𝜑    &   𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝑋)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)
 
Theoremsetrec1 42963 This is the first of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is closed under 𝐹. This effectively sets the actual value of setrecs(𝐹) as a lower bound for setrecs(𝐹), as it implies that any set generated by successive applications of 𝐹 is a member of 𝐵. This theorem "gets off the ground" because we can start by letting 𝐴 = ∅, and the hypotheses of the theorem will hold trivially.

Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon).

Proof summary: Assume that 𝐴𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 42961, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 42961.) Therefore, by setrec1lem4 42962, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4596, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 42956 for a detailed description of how the setrecs definition works.

(Contributed by Emmett Weisz, 9-Oct-2020.)

𝐵 = setrecs(𝐹)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)
 
Theoremsetrec2fun 42964* This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 42963 and setrec2v 42968 say that setrecs(𝐹) is the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7200) to the other class.

(Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   Fun 𝐹    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetrec2lem1 42965* Lemma for setrec2 42967. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
 
Theoremsetrec2lem2 42966* Lemma for setrec2 42967. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
 
Theoremsetrec2 42967* This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 42963 and setrec2v 42968 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7200) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetrec2v 42968* Version of setrec2 42967 with a dv condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetis 42969* Version of setrec2 42967 expressed as an induction schema. This theorem is a generalization of tfis3 7204. (Contributed by Emmett Weisz, 27-Feb-2022.)
𝐵 = setrecs(𝐹)    &   (𝑏 = 𝐴 → (𝜓𝜒))    &   (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))       (𝜑 → (𝐴𝐵𝜒))
 
20.37.2.2  Examples and properties of set recursion
 
Theoremelsetrecslem 42970* Lemma for elsetrecs 42971. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 42968. To see why this lemma also requires setrec1 42963, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
 
Theoremelsetrecs 42971* A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 42963 and setrec2 42967, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of are replaced by for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
 
Theoremsetrecsss 42972 The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐹𝐺)       (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))
 
Theoremsetrecsres 42973 A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.)
𝐵 = setrecs(𝐹)    &   (𝜑 → Fun 𝐹)       (𝜑𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵)))
 
Theoremvsetrec 42974 Construct V using set recursion. The proof indirectly uses trcl 8768, which relies on rec, but theoretically 𝐶 in trcl 8768 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.)
𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)       setrecs(𝐹) = V
 
Theorem0setrec 42975 If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.)
(𝜑 → (𝐹‘∅) = ∅)       (𝜑 → setrecs(𝐹) = ∅)
 
Theoremonsetreclem1 42976* Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝐹𝑎) = { 𝑎, suc 𝑎}
 
Theoremonsetreclem2 42977* Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
 
Theoremonsetreclem3 42978* Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
 
Theoremonsetrec 42979 Construct On using set recursion. When 𝑥 ∈ On, the function 𝐹 constructs the least ordinal greater than any of the elements of 𝑥, which is 𝑥 for a limit ordinal and suc 𝑥 for a successor ordinal.

For example, (𝐹‘{1𝑜, 2𝑜}) = { {1𝑜, 2𝑜}, suc {1𝑜, 2𝑜}} = {2𝑜, 3𝑜} which contains 3𝑜, and (𝐹‘ω) = { ω, suc ω} = {ω, ω +𝑜 1𝑜}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated.

Any function 𝐹 fulfilling lemmas onsetreclem2 42977 and onsetreclem3 42978 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 32029.

The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.)

𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       setrecs(𝐹) = On
 
20.37.3  Construction of Games and Surreal Numbers

Model organization after organization of reals - see TOC

 
Syntaxcpg 42980 Extend class notation to include the class of partizan game forms.
class Pg
 
Definitiondf-pg 42981 Define the class of partizan games. More precisely, this is the class of partizan game forms, many of which represent equal partisan games. In metamath, equality between partizan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.)
Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
 
Theoremelpglem1 42982* Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
 
Theoremelpglem2 42983* Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.)
(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
 
Theoremelpglem3 42984* Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
 
Theoremelpg 42985 Membership in the class of partizan games. In ONAG this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.)
(𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
 
20.38  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

 
20.38.1  Natural deduction
 
Theorem19.8ad 42986 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 2206. (Contributed by DAW, 13-Feb-2017.)
(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsbidd 42987 An identity theorem for substitution. See sbid 2270. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
(𝜑 → [𝑥 / 𝑥]𝜓)       (𝜑𝜓)
 
Theoremsbidd-misc 42988 An identity theorem for substitution. See sbid 2270. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑𝜓))
 
20.38.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

 
Syntaxcge-real 42989 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 42991.
class
 
Syntaxcgt 42990 Extend wff notation to include the 'greater than' relation, see df-gt 42992.
class >
 
Definitiondf-gte 42991 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 10282.

We do not write this as (𝑥𝑦𝑦𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦𝑦 < 𝑥), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: > = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦 < 𝑥)} and ≥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦𝑥)} but these are very complicated. This definition of , and the similar one for > (df-gt 42992), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 42993 for a more conventional expression of the relationship between < and >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

≥ =
 
Definitiondf-gt 42992 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 10151. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 42991 for a discussion on why this approach is used for the definition. See gt-lt 42994 and gt-lth 42996 for more conventional expression of the relationship between < and >.

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

> = <
 
Theoremgte-lte 42993 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
 
Theoremgt-lt 42994 Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
 
Theoremgte-lteh 42995 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵𝐵𝐴)
 
Theoremgt-lth 42996 Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 > 𝐵𝐵 < 𝐴)
 
Theoremex-gt 42997 Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
¬ 0 > 0
 
Theoremex-gte 42998 Simple example of , in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
0 ≥ 0
 
20.38.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 42999 Extend class notation to include the hyperbolic sine function, see df-sinh 43002.
class sinh
 
Syntaxccosh 43000 Extend class notation to include the hyperbolic cosine function. see df-cosh 43003.
class cosh
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