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Theorem elqsn0 6932
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 2404 . 2  |-  ( A /. R )  =  ( A /. R
)
2 neeq1 2575 . 2  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =/=  (/) 
<->  B  =/=  (/) ) )
3 eleq2 2465 . . . 4  |-  ( dom 
R  =  A  -> 
( x  e.  dom  R  <-> 
x  e.  A ) )
43biimpar 472 . . 3  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  x  e.  dom  R )
5 ecdmn0 6906 . . 3  |-  ( x  e.  dom  R  <->  [ x ] R  =/=  (/) )
64, 5sylib 189 . 2  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  [ x ] R  =/=  (/) )
71, 2, 6ectocld 6930 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   (/)c0 3588   dom cdm 4837   [cec 6862   /.cqs 6863
This theorem is referenced by:  ecelqsdm  6933  0nsr  8910  sylow1lem3  15189  vitalilem5  19457  prtlem400  26609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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