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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

Proof of Theorem elqsn0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . 2
2 neeq1 2609 . 2
3 eleq2 2497 . . . 4
43biimpar 472 . . 3
5 ecdmn0 6947 . . 3
64, 5sylib 189 . 2
71, 2, 6ectocld 6971 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   wne 2599  c0 3628   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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