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Theorem ecexr 6912
Description: A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr  |-  ( A  e.  [ B ] R  ->  B  e.  _V )

Proof of Theorem ecexr
StepHypRef Expression
1 n0i 3635 . . 3  |-  ( A  e.  ( R " { B } )  ->  -.  ( R " { B } )  =  (/) )
2 snprc 3873 . . . . 5  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
3 imaeq2 5201 . . . . 5  |-  ( { B }  =  (/)  ->  ( R " { B } )  =  ( R " (/) ) )
42, 3sylbi 189 . . . 4  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  ( R " (/) ) )
5 ima0 5223 . . . 4  |-  ( R
" (/) )  =  (/)
64, 5syl6eq 2486 . . 3  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  (/) )
71, 6nsyl2 122 . 2  |-  ( A  e.  ( R " { B } )  ->  B  e.  _V )
8 df-ec 6909 . 2  |-  [ B ] R  =  ( R " { B }
)
97, 8eleq2s 2530 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {csn 3816   "cima 4883   [cec 6905
This theorem is referenced by:  relelec  6947  ecdmn0  6949  erdisj  6954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-ec 6909
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