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Theorem elecg 8692
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
elecg ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Proof of Theorem elecg
StepHypRef Expression
1 elimasng 6041 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
21ancoms 460 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3 df-ec 8651 . . 3 [𝐵]𝑅 = (𝑅 “ {𝐵})
43eleq2i 2830 . 2 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ (𝑅 “ {𝐵}))
5 df-br 5107 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
62, 4, 53bitr4g 314 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  {csn 4587  cop 4593   class class class wbr 5106  cima 5637  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651
This theorem is referenced by:  elec  8693  relelec  8694  ecdmn0  8696  erth  8698  erdisj  8701  qsel  8736  orbsta  19094  sylow2alem1  19400  sylow2blem1  19403  sylow3lem3  19412  efgi2  19508  tgpconncompeqg  23466  xmetec  23790  blpnfctr  23792  xmetresbl  23793  xrsblre  24177  ecref  31628  ecxpid  32151  lsmsnorb  32175  ghmquskerlem1  32198  ecin0  36816  eqvrelth  37076  qsalrel  40667
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