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Theorem elec 8692
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
elec.1 𝐴 ∈ V
elec.2 𝐵 ∈ V
Assertion
Ref Expression
elec (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)

Proof of Theorem elec
StepHypRef Expression
1 elec.1 . 2 𝐴 ∈ V
2 elec.2 . 2 𝐵 ∈ V
3 elecg 8691 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
41, 2, 3mp2an 690 1 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2106  Vcvv 3445   class class class wbr 5105  [cec 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8650
This theorem is referenced by:  ecid  8721  sylow2alem2  19400  sylow2a  19401  sylow2blem1  19402  efgval2  19506  efgrelexlemb  19532  efgcpbllemb  19537  frgpnabllem1  19651  tgpconncomp  23464  qustgphaus  23474  vitalilem2  24973  vitalilem3  24974  isbndx  36241  prtlem10  37327  prtlem19  37340  prter3  37344
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