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Theorem elqsn0 6498
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 elqsn0m 6497 . 2  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
2 n0r 3376 . 2  |-  ( E. x  x  e.  B  ->  B  =/=  (/) )
31, 2syl 14 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480    =/= wne 2308   (/)c0 3363   dom cdm 4539   /.cqs 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by:  0nnq  7179  0nsr  7564
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