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Theorem elec 8325
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 8324 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
41, 2, 3mp2an 691 1 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  Vcvv 3480   class class class wbr 5053  [cec 8279
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-xp 5549  df-cnv 5551  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ec 8283
This theorem is referenced by:  ecid  8354  sylow2alem2  18741  sylow2a  18742  sylow2blem1  18743  efgval2  18848  efgrelexlemb  18874  efgcpbllemb  18879  frgpnabllem1  18991  tgpconncomp  22716  qustgphaus  22726  vitalilem2  24211  vitalilem3  24212  isbndx  35132  prtlem10  36073  prtlem19  36086  prter3  36090
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