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Theorem elec 7673
 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 7672 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
41, 2, 3mp2an 704 1 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  [cec 7627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631 This theorem is referenced by:  ecid  7699  sylow2alem2  17856  sylow2a  17857  sylow2blem1  17858  efgval2  17960  efgrelexlemb  17986  efgcpbllemb  17991  frgpnabllem1  18099  tgpconcomp  21726  qustgphaus  21736  vitalilem2  23184  vitalilem3  23185  isbndx  32751  prtlem10  33168  prtlem19  33181  prter3  33185
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