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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevengpoap3 38701* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
 
Theoremnnsum4primeseven 38702* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 4 ) ) N  =  sum_ k  e.  (
 1 ... 4 ) ( f `  k ) ) )
 
Theoremnnsum4primesevenALTV 38703* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 4
 ) ) N  =  sum_
 k  e.  ( 1
 ... 4 ) ( f `  k ) ) )
 
Theoremwtgoldbnnsum4prm 38704* If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  4  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theoremstgoldbnnsum4prm 38705* If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  A. n  e.  ( ZZ>=
 `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  n  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) ) )
 
Theorembgoldbnnsum3prm 38706* If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)
 |-  ( A. m  e. Even  ( 4  <  m  ->  m  e. GoldbachEven 
 )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  3  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theorembgoldbtbndlem1 38707 Lemma 1 for bgoldbtbnd 38711: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  e.  ( 7 [,); 1 3 ) ) 
 ->  N  e. GoldbachOddALTV  )
 
Theorembgoldbtbndlem2 38708* Lemma 2 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  S  =  ( X  -  ( F `  ( I  -  1 ) ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  (
 1..^ D ) ) 
 ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 ) ) ) 
 /\  ( X  -  ( F `  I ) )  <_  4 )  ->  ( S  e. Even  /\  S  <  N  /\  4  <  S ) ) )
 
Theorembgoldbtbndlem3 38709* Lemma 3 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   &    |-  S  =  ( X  -  ( F `  I ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  ( 1..^ D ) )  ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 )
 ) )  /\  4  <  S )  ->  ( S  e. Even  /\  S  <  N 
 /\  4  <  S ) ) )
 
Theorembgoldbtbndlem4 38710* Lemma 4 for bgoldbtbnd 38711. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ( ( ph  /\  I  e.  ( 1..^ D ) )  /\  X  e. Odd  )  ->  ( ( X  e.  (
 ( F `  I
 ) [,) ( F `  ( I  +  1
 ) ) )  /\  ( X  -  ( F `  I ) ) 
 <_  4 )  ->  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  (
 ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
 )  /\  X  =  ( ( p  +  q )  +  r
 ) ) ) )
 
Theorembgoldbtbnd 38711* If the binary Goldbach conjecture is valid up to an integer  N, and there is a series ("ladder") of primes with a difference of at most  N up to an integer  M, then the strong ternary Goldbach conjecture is valid up to  M, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ph  ->  A. n  e. Odd  ( ( 7  < 
 n  /\  n  <  M )  ->  n  e. GoldbachOddALTV  )
 )
 
Axiomax-bgbltosilva 38712 The binary Goldbach conjecture is valid for all even numbers less than or equal to 4 x 10^18, see result of [OeSilva] p. ?. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.)
 |-  (
 ( N  e. Even  /\  4  <  N  /\  N  <_  ( 4  x.  ( 10
 ^; 1 8 ) ) )  ->  N  e. GoldbachEven  )
 
Theorembgoldbachlt 38713* The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big  m). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 38712. (Contributed by AV, 3-Aug-2020.)
 |-  E. m  e.  NN  ( ( 4  x.  ( 10 ^; 1 8 ) )  <_  m  /\  A. n  e. Even  (
 ( 4  <  n  /\  n  <  m ) 
 ->  n  e. GoldbachEven  ) )
 
Axiomax-hgprmladder 38714 There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.)
 |-  E. d  e.  ( ZZ>= `  3 ) E. f  e.  (RePart `  d ) ( ( ( f `  0
 )  =  7  /\  ( f `  1
 )  = ; 1 3  /\  (
 f `  d )  =  (; 8 9  x.  ( 10 ^; 2 9 ) ) )  /\  A. i  e.  ( 0..^ d ) ( ( f `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( f `  ( i  +  1 ) )  -  ( f `  i ) )  < 
 ( ( 4  x.  ( 10 ^; 1 8 ) )  -  4 )  /\  4  <  ( ( f `
  ( i  +  1 ) )  -  ( f `  i
 ) ) ) )
 
Theoremtgblthelfgott 38715 The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 38713, ax-hgprmladder 38714 and bgoldbtbnd 38711. (Contributed by AV, 4-Aug-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  <  (; 8 8  x.  ( 10 ^; 2 9 ) ) )  ->  N  e. GoldbachOddALTV  )
 
Theoremtgoldbachlt 38716* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big  m greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 38715. (Contributed by AV, 4-Aug-2020.)
 |-  E. m  e.  NN  ( ( 8  x.  ( 10 ^; 3 0 ) )  <  m  /\  A. n  e. Odd  (
 ( 7  <  n  /\  n  <  m ) 
 ->  n  e. GoldbachOddALTV  ) )
 
Axiomax-tgoldbachgt 38717* The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed  m). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for m = 10^27. Temporarily provided as "axiom". (Contributed by AV, 2-Aug-2020.)
 |-  E. m  e.  NN  ( m  <_  ( 10 ^; 2 7 )  /\  A. n  e. Odd  ( m  <  n  ->  n  e. GoldbachOddALTV  ) )
 
Theoremtgoldbach 38718 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 38716 and ax-tgoldbachgt 38717. (Contributed by AV, 2-Aug-2020.)
 |-  A. n  e. Odd  ( 7  <  n  ->  n  e. GoldbachOddALTV  )
 
21.33.5  Proth's theorem
 
Theoremmodexp2m1d 38719 The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  1  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( -u 1  mod  E ) )   =>    |-  ( ph  ->  ( ( A ^ 2 )  mod  E )  =  1 )
 
Theoremproththdlem 38720 Lemma for proththd 38721. (Contributed by AV, 4-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   =>    |-  ( ph  ->  ( P  e.  NN  /\  1  <  P  /\  (
 ( P  -  1
 )  /  2 )  e.  NN ) )
 
Theoremproththd 38721* Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 14786), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   &    |-  ( ph  ->  K  <  (
 2 ^ N ) )   &    |-  ( ph  ->  E. x  e.  ZZ  (
 ( x ^ (
 ( P  -  1
 )  /  2 )
 )  mod  P )  =  ( -u 1  mod  P ) )   =>    |-  ( ph  ->  P  e.  Prime )
 
Theorem5tcu2e40 38722 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
 |-  (
 5  x.  ( 2 ^ 3 ) )  = ; 4 0
 
Theorem3exp4mod41 38723 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
 |-  (
 ( 3 ^ 4
 )  mod ; 4 1 )  =  ( -u 1  mod ; 4 1 )
 
Theorem41prothprmlem1 38724 Lemma 1 for 41prothprm 38726. (Contributed by AV, 4-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( P  -  1 )  /  2
 )  = ; 2 0
 
Theorem41prothprmlem2 38725 Lemma 2 for 41prothprm 38726. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( 3 ^
 ( ( P  -  1 )  /  2
 ) )  mod  P )  =  ( -u 1  mod  P )
 
Theorem41prothprm 38726 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( P  =  ( ( 5  x.  (
 2 ^ 3 ) )  +  1 ) 
 /\  P  e.  Prime )
 
21.33.6  Words over a set (extension)
 
21.33.6.1  Last symbol of a word (extension)
 
Theoremlswn0 38727 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( (/) is the last symbol) and invalid cases ( (/) means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  (/)  e/  V  /\  ( # `  W )  =/=  0 )  ->  ( lastS  `  W )  =/=  (/) )
 
21.33.6.2  Concatenations with singleton words (extension)
 
Theoremccatw2s1cl 38728 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  (
 ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W ++  <" X "> ) ++  <" Y "> )  e. Word  V )
 
21.33.6.3  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix), or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

 
Syntaxcpfx 38729 Syntax for the prefix operator.
 class prefix
 
Definitiondf-pfx 38730* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
 |- prefix  =  ( s  e.  _V ,  l  e.  NN0  |->  ( s substr  <. 0 ,  l >. ) )
 
Theorempfxval 38731 Value of a prefix. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e.  V  /\  L  e.  NN0 )  ->  ( S prefix  L )  =  ( S substr  <. 0 ,  L >. ) )
 
Theorempfx00 38732 A zero length prefix. (Contributed by AV, 2-May-2020.)
 |-  ( S prefix  0 )  =  (/)
 
Theorempfx0 38733 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
 |-  ( (/) prefix  L )  =  (/)
 
Theorempfxcl 38734 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  L )  e. Word  A )
 
Theorempfxmpt 38735* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( x  e.  (
 0..^ L )  |->  ( S `  x ) ) )
 
Theorempfxres 38736 Value of the prefix extractor as restriction. Could replace swrd0val 12716. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( S  |`  ( 0..^ L ) ) )
 
Theorempfxf 38737 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 12722. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ( W prefix  L ) : ( 0..^ L ) --> V )
 
Theorempfxfn 38738 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  V  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  Fn  ( 0..^ L ) )
 
Theorempfxlen 38739 Length of a prefix. Could replace swrd0len 12717. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( # `  ( S prefix  L ) )  =  L )
 
Theorempfxid 38740 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  ( # `  S ) )  =  S )
 
Theorempfxrn 38741 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ran  ( W prefix  L )  C_  V )
 
Theorempfxn0 38742 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12725. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN  /\  L  <_  ( # `  W ) )  ->  ( W prefix  L )  =/=  (/) )
 
Theorempfxnd 38743 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  <  L )  ->  ( W prefix  L )  =  (/) )
 
Theorempfxlen0 38744 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12731. TODO-AV: Really useful? swrd0len0 12731 is only used in wwlknred 25386. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  =  ( L  +  1 ) )  ->  ( # `
  ( W prefix  L ) )  =  L )
 
Theoremaddlenrevpfx 38745 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W substr 
 <. M ,  ( # `  W ) >. ) )  +  ( # `  ( W prefix  M ) ) )  =  ( # `  W ) )
 
Theoremaddlenpfx 38746 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W prefix  M ) )  +  ( # `  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theorempfxfv 38747 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12734. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) )  /\  I  e.  ( 0..^ L ) )  ->  ( ( W prefix  L ) `  I
 )  =  ( W `
  I ) )
 
Theorempfxfv0 38748 The first symbol in a prefix of a word. Could replace swrd0fv0 12735. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( ( W prefix  L ) `  0 )  =  ( W `  0
 ) )
 
Theorempfxtrcfv 38749 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12736. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/)  /\  I  e.  ( 0..^ ( ( # `  W )  -  1 ) ) ) 
 ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  I )  =  ( W `  I
 ) )
 
Theorempfxtrcfv0 38750 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12737. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  0 )  =  ( W `  0
 ) )
 
Theorempfxfvlsw 38751 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12738. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( lastS  `  ( W prefix  L ) )  =  ( W `  ( L  -  1 ) ) )
 
Theorempfxeq 38752* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12739. (Contributed by AV, 4-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `
  W )  /\  N  <_  ( # `  U ) ) )  ->  ( ( W prefix  M )  =  ( U prefix  N )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
 )  =  ( U `
  i ) ) ) )
 
Theorempfxtrcfvl 38753 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12745. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( lastS  `  ( W prefix  (
 ( # `  W )  -  1 ) ) )  =  ( W `
  ( ( # `  W )  -  2
 ) ) )
 
Theorempfxsuffeqwrdeq 38754 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12748. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  ( # `  S )  /\  ( ( W prefix  I
 )  =  ( S prefix  I )  /\  ( W substr  <. I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W )
 >. ) ) ) ) )
 
Theorempfxsuff1eqwrdeq 38755 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12749. (Contributed by AV, 6-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  ( # `  U )  /\  ( ( W prefix  (
 ( # `  W )  -  1 ) )  =  ( U prefix  (
 ( # `  W )  -  1 ) ) 
 /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
 
Theoremdisjwrdpfx 38756* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12750. (Contributed by AV, 6-May-2020.)
 |- Disj  y  e.  W  { x  e. Word  V  |  ( x prefix  N )  =  y }
 
Theoremccatpfx 38757 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
 |-  (
 ( S  e. Word  A  /\  Y  e.  ( 0
 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) )  ->  ( ( S prefix  Y ) ++  ( S substr  <. Y ,  Z >. ) )  =  ( S prefix  Z )
 )
 
Theorempfxccat1 38758 Recover the left half of a concatenated word. Could replace swrdccat1 12752. (Contributed by AV, 6-May-2020.)
 |-  (
 ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T ) prefix  ( # `  S ) )  =  S )
 
Theorempfx1 38759 A prefix of length 1. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W prefix  1 )  =  <" ( W `
  0 ) "> )
 
Theorempfx2 38760 A prefix of length 2. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( W prefix  2 )  =  <" ( W `
  0 ) ( W `  1 ) "> )
 
Theorempfxswrd 38761 A prefix of a subword. Could replace swrd0swrd 12756. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( L  e.  ( 0 ... ( N  -  M ) )  ->  ( ( W substr  <. M ,  N >. ) prefix  L )  =  ( W substr  <. M ,  ( M  +  L ) >. ) ) )
 
Theoremswrdpfx 38762 A subword of a prefix. Could replace swrdswrd0 12757. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) ) 
 ->  ( ( W prefix  N ) substr 
 <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )
 
Theorempfxpfx 38763 A prefix of a prefix. Could replace swrd0swrd0 12758. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  L  e.  ( 0 ... N ) )  ->  ( ( W prefix  N ) prefix  L )  =  ( W prefix  L ) )
 
Theorempfxpfxid 38764 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 12759. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  N ) prefix  N )  =  ( W prefix  N ) )
 
Theorempfxcctswrd 38765 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 12760. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W )
 >. ) )  =  W )
 
Theoremlenpfxcctswrd 38766 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 12761. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theoremlenrevpfxcctswrd 38767 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 12762. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W substr  <. M ,  ( # `
  W ) >. ) ++  ( W prefix  M )
 ) )  =  ( # `  W ) )
 
Theorempfxlswccat 38768 Reconstruct a nonempty word from its prefix and last symbol. Could replace wrdeqcats1OLD 12769 resp. swrdccatwrd 12763. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) ++ 
 <" ( lastS  `  W ) "> )  =  W )
 
Theoremccats1pfxeq 38769 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 12764. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  U  =  ( W ++ 
 <" ( lastS  `  U ) "> ) ) )
 
Theoremccats1pfxeqrex 38770* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 12774. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  E. s  e.  V  U  =  ( W ++  <" s "> ) ) )
 
Theorempfxccatin12lem1 38771 Lemma 1 for pfxccatin12 38773. Could replace swrdccatin12lem2b 12781. (Contributed by AV, 9-May-2020.)
 |-  (
 ( M  e.  (
 0 ... L )  /\  N  e.  ( L ... X ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  -  ( L  -  M ) )  e.  ( 0..^ ( N  -  L ) ) ) )
 
Theorempfxccatin12lem2 38772 Lemma 2 for pfxccatin12 38773. Could replace swrdccatin12lem2 12784. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
 ( A ++  B ) substr  <. M ,  N >. ) `
  K )  =  ( ( B prefix  ( N  -  L ) ) `
  ( K  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) ) )
 
Theorempfxccatin12 38773 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12786. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) ) )
 
Theorempfxccat3 38774 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12787. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 N )  /\  N  e.  ( 0 ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. M ,  N >. ) ,  if ( L 
 <_  M ,  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) ++  ( B prefix 
 ( N  -  L ) ) ) ) ) ) )
 
Theorempfxccatpfx1 38775 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( 0
 ... L ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A prefix  N ) )
 
Theorempfxccatpfx2 38776 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( ( L  +  1 )
 ... ( L  +  M ) ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A ++  ( B prefix  ( N  -  L ) ) ) )
 
Theorempfxccat3a 38777 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12789. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( L  +  M )
 )  ->  ( ( A ++  B ) prefix  N )  =  if ( N 
 <_  L ,  ( A prefix  N ) ,  ( A ++  ( B prefix  ( N  -  L ) ) ) ) ) )
 
Theorempfxccatid 38778 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12792. (Contributed by AV, 10-May-2020.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A ) )  ->  ( ( A ++  B ) prefix  N )  =  A )
 
Theoremccats1pfxeqbi 38779 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12793. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  <->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
 
Theorempfxccatin12d 38780 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12796. (Contributed by AV, 10-May-2020.)
 |-  ( ph  ->  ( # `  A )  =  L )   &    |-  ( ph  ->  ( A  e. Word  V 
 /\  B  e. Word  V ) )   &    |-  ( ph  ->  M  e.  ( 0 ...
 L ) )   &    |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )   =>    |-  ( ph  ->  ( ( A ++  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) )
 
Theoremreuccatpfxs1lem 38781* Lemma for reuccatpfxs1 38782. Could replace reuccats1lem 12775. (Contributed by AV, 9-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e.  X )  /\  A. s  e.  V  ( ( W ++ 
 <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  ->  U  =  ( W ++  <" S "> ) ) )
 
Theoremreuccatpfxs1 38782* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 12776. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( E! v  e.  V  ( W ++  <" v "> )  e.  X  ->  E! w  e.  X  W  =  ( w prefix  ( # `  W ) ) ) )
 
Theoremsplvalpfx 38783 Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  ( F  e.  NN0  /\  T  e.  X  /\  R  e.  Y )
 )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S prefix  F ) ++  R ) ++  ( S substr  <. T ,  ( # `  S ) >. ) ) )
 
Theoremrepswpfx 38784 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  ->  ( ( S repeatS  N ) prefix  L )  =  ( S repeatS  L ) )
 
Theoremcshword2 38785 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( ( W substr  <. ( N 
 mod  ( # `  W ) ) ,  ( # `
  W ) >. ) ++  ( W prefix  ( N  mod  ( # `  W ) ) ) ) )
 
Theorempfxco 38786 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  A  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  F : A
 --> B )  ->  ( F  o.  ( W prefix  N ) )  =  (
 ( F  o.  W ) prefix  N ) )
 
21.33.7  Auxiliary theorems for graph theory

Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

 
21.33.7.1  Negated equality and membership - extension
 
Theoremelnelall 38787 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  e.  B  ->  ( A  e/  B  ->  ph ) )
 
21.33.7.2  Subclasses and subsets - extension
 
Theoremclel5 38788* Alternate definition of class membership: a class  X is an element of another class  A iff there is an element of  A equal to  X. (Contributed by AV, 13-Nov-2020.)
 |-  ( X  e.  A  <->  E. x  e.  A  X  =  x )
 
Theoremdfss7 38789* Alternate definition of subclass relationship: a class  A is a subclass of another class  B iff each element of  A is equal to an element of  B. (Contributed by AV, 13-Nov-2020.)
 |-  ( A  C_  B  <->  A. x  e.  A  E. y  e.  B  x  =  y )
 
21.33.7.3  The empty set - extension
 
Theoremralnralall 38790* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
 |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
 /\  A. x  e.  A  -.  ph )  ->  ps )
 )
 
Theoremfalseral0 38791* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
 |-  (
 ( A. x  -.  ph  /\ 
 A. x  e.  A  ph )  ->  A  =  (/) )
 
Theoremralralimp 38792* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
 |-  (
 ( ph  /\  A  =/=  (/) )  ->  ( A. x  e.  A  (
 ( ph  ->  ( th  \/  ta ) )  /\  -. 
 th )  ->  ta )
 )
 
21.33.7.4  Unordered and ordered pairs - extension
 
Theoremelpwdifsn 38793 A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)
 |-  (
 ( S  e.  W  /\  S  C_  V  /\  A  e/  S )  ->  S  e.  ~P ( V  \  { A }
 ) )
 
Theorempr1eqbg 38794 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =  C 
 <->  { A ,  B }  =  { B ,  C } ) )
 
Theorempr1nebg 38795 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =/=  C  <->  { A ,  B }  =/=  { B ,  C } ) )
 
Theoremprelpw 38796 A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A  e.  C  /\  B  e.  C ) 
 <->  { A ,  B }  e.  ~P C ) )
 
Theoremrexdifpr 38797 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
 |-  ( E. x  e.  ( A  \  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/=  C  /\  ph )
 )
 
Theoremissn 38798* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
 |-  ( E. x  e.  A  A. y  e.  A  x  =  y  ->  E. z  A  =  { z } )
 
Theoremn0snor2el 38799* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  E. y  e.  A  x  =/=  y  \/  E. z  A  =  { z } )
 )
 
Theoremopidg 38800 The ordered pair  <. A ,  A >. in Kuratowski's representation. Closed form of opid 4142. (Contributed by AV, 18-Sep-2020.)
 |-  ( A  e.  _V  ->  <. A ,  A >.  =  { { A } } )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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