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Theorem s1val 11744
Description: Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1val  |-  ( A  e.  V  ->  <" A ">  =  { <. 0 ,  A >. } )

Proof of Theorem s1val
StepHypRef Expression
1 df-s1 11717 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
2 fvi 5775 . . . 4  |-  ( A  e.  V  ->  (  _I  `  A )  =  A )
32opeq2d 3983 . . 3  |-  ( A  e.  V  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  A >. )
43sneqd 3819 . 2  |-  ( A  e.  V  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  A >. } )
51, 4syl5eq 2479 1  |-  ( A  e.  V  ->  <" A ">  =  { <. 0 ,  A >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {csn 3806   <.cop 3809    _I cid 4485   ` cfv 5446   0cc0 8982   <"cs1 11711
This theorem is referenced by:  s1cl  11747  s1fv  11752  s111  11754  wrdexb  11755  s1co  11794  s2prop  11853  gsumws1  14777
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-s1 11717
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