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Theorem ecexr 7217
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 3751 . . 3  |-  ( A  e.  ( R " { B } )  ->  -.  ( R " { B } )  =  (/) )
2 snprc 4048 . . . . 5  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
3 imaeq2 5274 . . . . 5  |-  ( { B }  =  (/)  ->  ( R " { B } )  =  ( R " (/) ) )
42, 3sylbi 195 . . . 4  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  ( R " (/) ) )
5 ima0 5293 . . . 4  |-  ( R
" (/) )  =  (/)
64, 5syl6eq 2511 . . 3  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  (/) )
71, 6nsyl2 127 . 2  |-  ( A  e.  ( R " { B } )  ->  B  e.  _V )
8 df-ec 7214 . 2  |-  [ B ] R  =  ( R " { B }
)
97, 8eleq2s 2562 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746   {csn 3986   "cima 4952   [cec 7210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-cnv 4957  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-ec 7214
This theorem is referenced by:  relelec  7252  ecdmn0  7254  erdisj  7259
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