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Theorem ecexr 7314
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 3772 . . 3  |-  ( A  e.  ( R " { B } )  ->  -.  ( R " { B } )  =  (/) )
2 snprc 4074 . . . . 5  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
3 imaeq2 5319 . . . . 5  |-  ( { B }  =  (/)  ->  ( R " { B } )  =  ( R " (/) ) )
42, 3sylbi 195 . . . 4  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  ( R " (/) ) )
5 ima0 5338 . . . 4  |-  ( R
" (/) )  =  (/)
64, 5syl6eq 2498 . . 3  |-  ( -.  B  e.  _V  ->  ( R " { B } )  =  (/) )
71, 6nsyl2 127 . 2  |-  ( A  e.  ( R " { B } )  ->  B  e.  _V )
8 df-ec 7311 . 2  |-  [ B ] R  =  ( R " { B }
)
97, 8eleq2s 2549 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1381    e. wcel 1802   _Vcvv 3093   (/)c0 3767   {csn 4010   "cima 4988   [cec 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-ec 7311
This theorem is referenced by:  relelec  7350  ecdmn0  7352  erdisj  7357
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