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Theorem s111 11690
Description: The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s111  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  S  =  T
) )

Proof of Theorem s111
StepHypRef Expression
1 s1val 11680 . . 3  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 s1val 11680 . . 3  |-  ( T  e.  A  ->  <" T ">  =  { <. 0 ,  T >. } )
31, 2eqeqan12d 2403 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  { <. 0 ,  S >. }  =  { <. 0 ,  T >. } ) )
4 opex 4369 . . 3  |-  <. 0 ,  S >.  e.  _V
5 sneqbg 3912 . . 3  |-  ( <.
0 ,  S >.  e. 
_V  ->  ( { <. 0 ,  S >. }  =  { <. 0 ,  T >. }  <->  <. 0 ,  S >.  =  <. 0 ,  T >. ) )
64, 5mp1i 12 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( { <. 0 ,  S >. }  =  { <. 0 ,  T >. }  <->  <. 0 ,  S >.  = 
<. 0 ,  T >. ) )
7 0z 10226 . . . 4  |-  0  e.  ZZ
8 eqid 2388 . . . . 5  |-  0  =  0
9 opthg 4378 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
( 0  =  0  /\  S  =  T ) ) )
109baibd 876 . . . . 5  |-  ( ( ( 0  e.  ZZ  /\  S  e.  A )  /\  0  =  0 )  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
118, 10mpan2 653 . . . 4  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
127, 11mpan 652 . . 3  |-  ( S  e.  A  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
1312adantr 452 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
143, 6, 133bitrd 271 1  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  S  =  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758   <.cop 3761   0cc0 8924   ZZcz 10215   <"cs1 11647
This theorem is referenced by:  efgredlemc  15305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-i2m1 8992  ax-1ne0 8993  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-neg 9227  df-z 10216  df-s1 11653
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