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Theorem elqsi 6534
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 6532 . 2 (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
21ibi 175 1 (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  wrex 2436  [cec 6480   / cqs 6481
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-qs 6488
This theorem is referenced by:  ectocld  6548  ecoptocl  6569  eroveu  6573  dmaddpqlem  7299  nqpi  7300  nq0nn  7364
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