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Theorem ecelqsi 8755
Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ecelqsi.1 𝑅 ∈ V
Assertion
Ref Expression
ecelqsi (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Proof of Theorem ecelqsi
StepHypRef Expression
1 ecelqsi.1 . 2 𝑅 ∈ V
2 ecelqsw 8754 . 2 ((𝑅 ∈ V ∧ 𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
31, 2mpan 702 1 (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  Vcvv 3457  [cec 8680   / cqs 8681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395  ax-un 7722
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ec 8684  df-qs 8688
This theorem is referenced by:  ecopqsi  8756  addsrpr  11048  mulsrpr  11049  0r  11053  1sr  11054  m1r  11055  addclsr  11056  mulclsr  11057  quseccl0  19247  ghmqusnsglem1  19341  ghmquskerlem1  19344  ghmquskerco  19345  ghmqusker  19348  orbsta  19374  frgpeccl  19822  rngqiprngimf  21399  qsidomlem1  21440  qustgphaus  24241  vitalilem2  25729  vitalilem3  25730  rloccring  33504  rloc0g  33505  rloc1r  33506  rlocf1  33507  rlocinvunit  33508  rlocisunit  33509  fracfld  33544  nsgqusf1olem1  33638  qsdrngilem  33693  qsdrngi  33694  qsdrnglem2  33695  zringfrac  33761  pstmfval  34203
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