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Theorem elqsi 8643
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
elqsi (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem elqsi
StepHypRef Expression
1 elqsg 8641 . 2 (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
21ibi 267 1 (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3072  [cec 8580   / cqs 8581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3073  df-qs 8588
This theorem is referenced by:  ectocld  8657  ecoptocl  8680  eroveu  8685  nsgqusf1olem2  31992  qsidomlem2  32022  pstmxmet  32258
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