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Theorem elqsi 7970
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 7968 . 2 (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
21ibi 256 1 (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  wrex 3052  [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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-v 3343  df-qs 7920
This theorem is referenced by:  ectocld  7984  ecoptocl  8007  eroveu  8012  pstmxmet  30271
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