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Theorem s111 11448
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 11438 . . 3  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 s1val 11438 . . 3  |-  ( T  e.  A  ->  <" T ">  =  { <. 0 ,  T >. } )
31, 2eqeqan12d 2298 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  { <. 0 ,  S >. }  =  { <. 0 ,  T >. } ) )
4 opex 4237 . . 3  |-  <. 0 ,  S >.  e.  _V
5 sneqbg 3783 . . 3  |-  ( <.
0 ,  S >.  e. 
_V  ->  ( { <. 0 ,  S >. }  =  { <. 0 ,  T >. }  <->  <. 0 ,  S >.  =  <. 0 ,  T >. ) )
64, 5mp1i 11 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( { <. 0 ,  S >. }  =  { <. 0 ,  T >. }  <->  <. 0 ,  S >.  = 
<. 0 ,  T >. ) )
7 0z 10035 . . . 4  |-  0  e.  ZZ
8 eqid 2283 . . . . 5  |-  0  =  0
9 opthg 4246 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
( 0  =  0  /\  S  =  T ) ) )
109baibd 875 . . . . 5  |-  ( ( ( 0  e.  ZZ  /\  S  e.  A )  /\  0  =  0 )  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
118, 10mpan2 652 . . . 4  |-  ( ( 0  e.  ZZ  /\  S  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
127, 11mpan 651 . . 3  |-  ( S  e.  A  ->  ( <. 0 ,  S >.  = 
<. 0 ,  T >.  <-> 
S  =  T ) )
1312adantr 451 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <. 0 ,  S >.  =  <. 0 ,  T >.  <-> 
S  =  T ) )
143, 6, 133bitrd 270 1  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  S  =  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   0cc0 8737   ZZcz 10024   <"cs1 11405
This theorem is referenced by:  efgredlemc  15054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-neg 9040  df-z 10025  df-s1 11411
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