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Theorem s111 11754
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 11744 . . 3  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 s1val 11744 . . 3  |-  ( T  e.  A  ->  <" T ">  =  { <. 0 ,  T >. } )
31, 2eqeqan12d 2450 . 2  |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T ">  <->  { <. 0 ,  S >. }  =  { <. 0 ,  T >. } ) )
4 opex 4419 . . 3  |-  <. 0 ,  S >.  e.  _V
5 sneqbg 3961 . . 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 10285 . . . 4  |-  0  e.  ZZ
8 eqid 2435 . . . . 5  |-  0  =  0
9 opthg 4428 . . . . . 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 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   0cc0 8982   ZZcz 10274   <"cs1 11711
This theorem is referenced by:  efgredlemc  15369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-neg 9286  df-z 10275  df-s1 11717
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