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Theorem elqsn0 6728
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 2283 . 2  |-  ( A /. R )  =  ( A /. R
)
2 neeq1 2454 . 2  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =/=  (/) 
<->  B  =/=  (/) ) )
3 eleq2 2344 . . . 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 6702 . . 3  |-  ( x  e.  dom  R  <->  [ x ] R  =/=  (/) )
64, 5sylib 188 . 2  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  [ x ] R  =/=  (/) )
71, 2, 6ectocld 6726 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   dom cdm 4689   [cec 6658   /.cqs 6659
This theorem is referenced by:  ecelqsdm  6729  0nsr  8701  sylow1lem3  14911  vitalilem5  18967  prtlem400  26150
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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  df-qs 6666
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