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Theorem elqsn0 7986
Description: A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
elqsn0 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)

Proof of Theorem elqsn0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2761 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 neeq1 2995 . 2 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 eleq2 2829 . . . 4 (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅𝑥𝐴))
43biimpar 503 . . 3 ((dom 𝑅 = 𝐴𝑥𝐴) → 𝑥 ∈ dom 𝑅)
5 ecdmn0 7959 . . 3 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
64, 5sylib 208 . 2 ((dom 𝑅 = 𝐴𝑥𝐴) → [𝑥]𝑅 ≠ ∅)
71, 2, 6ectocld 7984 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wne 2933  c0 4059  dom cdm 5267  [cec 7912   / cqs 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ec 7916  df-qs 7920
This theorem is referenced by:  ecelqsdm  7987  0nsr  10113  sylow1lem3  18236  vitalilem5  23601  prtlem400  34678
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