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Theorem s111 12632
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 12619 . . 3  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 s1val 12619 . . 3  |-  ( T  e.  A  ->  <" T ">  =  { <. 0 ,  T >. } )
31, 2eqeqan12d 2480 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  { <. 0 ,  S >. }  =  { <. 0 ,  T >. } ) )
4 opex 4720 . . 3  |-  <. 0 ,  S >.  e.  _V
5 sneqbg 4202 . . 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 10896 . . . 4  |-  0  e.  ZZ
8 eqid 2457 . . . . 5  |-  0  =  0
9 opthg 4731 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
( 0  =  0  /\  S  =  T ) ) )
109baibd 909 . . . . 5  |-  ( ( ( 0  e.  ZZ  /\  S  e.  A )  /\  0  =  0 )  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
118, 10mpan2 671 . . . 4  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
127, 11mpan 670 . . 3  |-  ( S  e.  A  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
1312adantr 465 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
143, 6, 133bitrd 279 1  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  S  =  T
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038   0cc0 9509   ZZcz 10885   <"cs1 12541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-neg 9827  df-z 10886  df-s1 12549
This theorem is referenced by:  2swrd1eqwrdeq  12691  s2eq2seq  12894  2swrd2eqwrdeq  12903  efgredlemc  16890  mvhf1  29116  pfxsuff1eqwrdeq  32525
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