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Theorem elqsn0 6744
Description: A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
elqsn0  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )

Proof of Theorem elqsn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . 2  |-  ( A /. R )  =  ( A /. R
)
2 neeq1 2467 . 2  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =/=  (/) 
<->  B  =/=  (/) ) )
3 eleq2 2357 . . . 4  |-  ( dom 
R  =  A  -> 
( x  e.  dom  R  <-> 
x  e.  A ) )
43biimpar 471 . . 3  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  x  e.  dom  R )
5 ecdmn0 6718 . . 3  |-  ( x  e.  dom  R  <->  [ x ] R  =/=  (/) )
64, 5sylib 188 . 2  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  [ x ] R  =/=  (/) )
71, 2, 6ectocld 6742 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   dom cdm 4705   [cec 6674   /.cqs 6675
This theorem is referenced by:  ecelqsdm  6745  0nsr  8717  sylow1lem3  14927  vitalilem5  18983  prtlem400  26841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-qs 6682
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