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Theorem s1eq 11439
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 5525 . . . 4  |-  ( A  =  B  ->  (  _I  `  A )  =  (  _I  `  B
) )
21opeq2d 3803 . . 3  |-  ( A  =  B  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  (  _I  `  B
) >. )
32sneqd 3653 . 2  |-  ( A  =  B  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  (  _I  `  B )
>. } )
4 df-s1 11411 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
5 df-s1 11411 . 2  |-  <" B ">  =  { <. 0 ,  (  _I  `  B ) >. }
63, 4, 53eqtr4g 2340 1  |-  ( A  =  B  ->  <" A ">  =  <" B "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {csn 3640   <.cop 3643    _I cid 4304   ` cfv 5255   0cc0 8737   <"cs1 11405
This theorem is referenced by:  s1eqd  11440  wrdind  11477  revs1  11483  vrmdval  14479  frgpup3lem  15086  vdegp1ci  23910
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-s1 11411
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