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Theorem elec 7829
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 7828 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
41, 2, 3mp2an 708 1 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  Vcvv 3231   class class class wbr 4685  [cec 7785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ec 7789
This theorem is referenced by:  ecid  7855  sylow2alem2  18079  sylow2a  18080  sylow2blem1  18081  efgval2  18183  efgrelexlemb  18209  efgcpbllemb  18214  frgpnabllem1  18322  tgpconncomp  21963  qustgphaus  21973  vitalilem2  23423  vitalilem3  23424  isbndx  33711  prtlem10  34469  prtlem19  34482  prter3  34486
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