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Theorem elqsn0 6973
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 2436 . 2  |-  ( A /. R )  =  ( A /. R
)
2 neeq1 2609 . 2  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =/=  (/) 
<->  B  =/=  (/) ) )
3 eleq2 2497 . . . 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 6947 . . 3  |-  ( x  e.  dom  R  <->  [ x ] R  =/=  (/) )
64, 5sylib 189 . 2  |-  ( ( dom  R  =  A  /\  x  e.  A
)  ->  [ x ] R  =/=  (/) )
71, 2, 6ectocld 6971 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   dom cdm 4878   [cec 6903   /.cqs 6904
This theorem is referenced by:  ecelqsdm  6974  0nsr  8954  sylow1lem3  15234  vitalilem5  19504  prtlem400  26719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-ec 6907  df-qs 6911
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