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Theorem ecexr 7308
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 3788 . . 3  |-  ( A  e.  ( R " { B } )  ->  -.  ( R " { B } )  =  (/) )
2 snprc 4079 . . . . 5  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
3 imaeq2 5321 . . . . 5  |-  ( { B }  =  (/)  ->  ( R " { B } )  =  ( R " (/) ) )
42, 3sylbi 195 . . . 4  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  ( R " (/) ) )
5 ima0 5340 . . . 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 7305 . 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 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016   "cima 4991   [cec 7301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-ec 7305
This theorem is referenced by:  relelec  7344  ecdmn0  7346  erdisj  7351
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