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Theorem s1eq 11455
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1eq  |-  ( A  =  B  ->  <" A ">  =  <" B "> )

Proof of Theorem s1eq
StepHypRef Expression
1 fveq2 5541 . . . 4  |-  ( A  =  B  ->  (  _I  `  A )  =  (  _I  `  B
) )
21opeq2d 3819 . . 3  |-  ( A  =  B  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  (  _I  `  B
) >. )
32sneqd 3666 . 2  |-  ( A  =  B  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  (  _I  `  B )
>. } )
4 df-s1 11427 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
5 df-s1 11427 . 2  |-  <" B ">  =  { <. 0 ,  (  _I  `  B ) >. }
63, 4, 53eqtr4g 2353 1  |-  ( A  =  B  ->  <" A ">  =  <" B "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {csn 3653   <.cop 3656    _I cid 4320   ` cfv 5271   0cc0 8753   <"cs1 11421
This theorem is referenced by:  s1eqd  11456  wrdind  11493  revs1  11499  vrmdval  14495  frgpup3lem  15102  vdegp1ci  23925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-s1 11427
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