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Theorem ecexr 6665
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 3460 . . 3  |-  ( A  e.  ( R " { B } )  ->  -.  ( R " { B } )  =  (/) )
2 snprc 3695 . . . . 5  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
3 imaeq2 5008 . . . . 5  |-  ( { B }  =  (/)  ->  ( R " { B } )  =  ( R " (/) ) )
42, 3sylbi 187 . . . 4  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  ( R " (/) ) )
5 ima0 5030 . . . 4  |-  ( R
" (/) )  =  (/)
64, 5syl6eq 2331 . . 3  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  (/) )
71, 6nsyl2 119 . 2  |-  ( A  e.  ( R " { B } )  ->  B  e.  _V )
8 df-ec 6662 . 2  |-  [ B ] R  =  ( R " { B }
)
97, 8eleq2s 2375 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   "cima 4692   [cec 6658
This theorem is referenced by:  relelec  6700  ecdmn0  6702  erdisj  6707
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  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-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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